Thompson_Jordan_2011

Transcription

Searching for the Magnetic
Interactions in the Rare Earth
Pyrochlore Oxide Yb2Ti2O7
by
Jordan Thompson
A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Master of Science
in
Physics
Waterloo, Ontario, Canada, 2011
c Jordan Thompson 2011
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
ii
Abstract
Various experiments on Yb2 Ti2 O7 have shown evidence of strange magnetic behaviour
at low temperatures. Specific heat measurements on powder samples of Yb2 Ti2 O7 show
evidence of a sharp peak, indicating the occurence of a first order phase transition. Meanwhile, neutron scattering, Mössbauer absorption, and µSR measurements find no evidence
of long range order below the temperature of this phase transition, leaving the nature of the
low temperature phase a mystery. Quantifying the magnetic interactions in this material
should allow us to better understand the low temperature behaviour of this material. In
this study, we fit a symmetry allowed nearest-neighbour bilinear exchange model to quasielastic neutron scattering data collected well above the temperature of the experimentally
observed phase transition. This neutron scattering data shows evidence of rods of scattering intensity along the h111i crystallographic directions. Neutron scattering probes the
correlations between magnetic moments in a material, so fitting an interaction model to
the neutron scattering is equivalent to fitting the interactions to the magnetic correlations.
These correlations are driven by the interactions between the magnetic moments, so the
neutron scattering should give us direct access to the form of these interactions. Using this
method we successfully identify an anisotropic nearest-neighbour bilinear exchange model
that reproduces the experimentally observed quasi-elastic neutron scattering. With this
model we then proceed to compute real space correlation functions, finding that the rods of
neutron scattering arise from the presence of strong correlations along nearest-neighbour
chains. We also compute the bulk susceptibility and local susceptibility, obtaining very
good fits to experiment with no variation of the model determined from the neutron scattering. The success of these calculations provides a further independent confirmation of
the success of our interaction model in describing the magnetic interactions in Yb2 Ti2 O7 .
Finally, we present a brief summary of ongoing work based on our anisotropic exchange
model, including mean field calculations to determine the low temperature ground state of
this model and classical Monte Carlo simulations to study the phase transition present in
this model. We also discuss potential further studies of this and other models.
iii
Acknowledgements
First, I would like to acknowledge the persistence, patience, and encouragement of
my supervisor, Dr. Michel Gingras. I have learned a great deal from my time studying
with him, and will be forever grateful for this experience. I would also like to thank
Dr. Paul McClarty for his large contributions to this work, including the development of
the RPA neutron scattering code, mean field calculations of the local susceptibility, and
diagonalization of crystal field parameterizations. I would also like to thank him for his
many explanations of physical concepts and theoretical methods.
I would like to thank: Dr. Pawel Stasiak, for the use of his parallel tempering classical
Monte Carlo code, and his assistance in understanding its use, Behnam Javanparast, for
his assistance with finding the ground state of our models of Yb2 Ti2 O7 , Anson Wong for
many useful discussions on spin-waves and the Ewald summation technique for treating
the long-range magnetic dipolar interaction, and the rest of the Condensed Matter Theory
group/Quantum Matters group at the University of Waterloo for many interesting and
enlightening discussions and assistance with various aspects of my work.
I would like to thank Prof. Henrik Ronnow and Dr. Louis-Pierre Regnault for the
experimental neutron scattering data that made most of this work possible.
Finally I would like to dedicate this work to my family, especially my parents, for their
support and encouragement during my studies. I would also like to thank my sister for her
assistance in editing this work.
iv
Table of Contents
List of Tables
x
List of Figures
xiii
1 Introduction - An Experimental Overview of Yb2 Ti2 O7
1
1.1
Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
The Pyrochlore Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Rare Earth Pyrochlores . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Ytterbium Titanate - Yb2 Ti2 O7 . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Previous Work on Yb2 Ti2 O7 . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.6
Paramagnetic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . .
19
2 The Model Hamiltonian for Yb2 Ti2 O7
27
2.1
The Crystal Field of Yb2 Ti2 O7
. . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
Magnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
Symmetry Allowed Bilinear Exchange Interactions . . . . . . . . . .
29
2.2.2
Alternative Representations of the Symmetry Allowed Bilinear Exchange Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
The Long-Range Magnetic Dipolar Interaction . . . . . . . . . . . .
32
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.2.3
2.3
v
3 Magnetic Neutron Scattering
33
3.1
The Microscopic Origin of Neutron Scattering . . . . . . . . . . . . . . . .
33
3.2
The Relationship Between Quasi-elastic Neutron Scattering and the Qdependent Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . .
41
The Random Phase Approximation Formula for Quasi-elastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.3
3.4
4 Determining the Magnetic Interactions in Yb2 Ti2 O7 from Paramagnetic
Quasi-elastic Neutron Scattering
49
4.1
Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.1.1
The Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . .
50
4.1.2
The Effective Energy Function and Implementation of the Metropolis
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
The Curie-Weiss Temperature Constraint and Additional Refinement of Simulated Annealing Results . . . . . . . . . . . . . . . . .
53
4.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.1.3
5 Real Space Correlations
5.1
64
Computing the Real Space Correlation Function in the Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6 The Local Susceptibility
72
6.1
Definition of the Local Susceptibility . . . . . . . . . . . . . . . . . . . . .
72
6.2
Calculating the Local Susceptibility . . . . . . . . . . . . . . . . . . . . . .
75
6.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
vi
7 Ongoing and Prospective Future Research
83
7.1
Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
7.2
Mean Field Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
7.3
Quantum Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
7.4
Ground States the Full {Je } Phase Space for the Non-Ideal Local-Planar
Pyrochlore Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Rods of Scattering: The Pyrochlore Heisenberg Ferromagnet . . . . . . . .
89
7.5
8 Conclusions
92
APPENDICES
96
A Crystal Field Parameterizations
97
A.1 The Crystal Field of Hodges et al. . . . . . . . . . . . . . . . . . . . . . . .
97
A.2 The Crystal Field of Cao et al.
98
. . . . . . . . . . . . . . . . . . . . . . . .
B Stevens-operator Equivalents
102
C Symmetry Allowed Exchange Interactions on the Pyrochlore Lattice
104
C.1 Deriving the Symmetry Allowed Exchange Interactions . . . . . . . . . . . 104
C.2 Other Representations of the Symmetry Allowed Nearest Neighbour Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
D Actions of the Symmetry Operations of the Oh Point Group
118
E Long Range Dipolar Interactions
121
F The Magnetic Form Factor of Yb3+
127
G Fourier Transform of Nearest Neighbour Interactions on the Pyrochlore
Lattice
129
H Effective Spin-1/2 Hamiltonians
131
vii
I
Units of the Magnetic Susceptibility
References
136
138
viii
List of Tables
1.1
FCC lattice vectors and tetrahedral sublattice vectors. . . . . . . . . . . .
4
1.2
FCC reciprocal lattice vectors. . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Local coordinates for the four sublattice sites of the pyrochlore lattice.
7
4.1
Average values of the coupling parameters {Jn } resulting from simulated
annealing. The indicated uncertainties correspond to one standard deviation. 56
4.2
Average values of the coupling parameters {Je }. . . . . . . . . . . . . . . .
4.3
Selected values of {Je } for comparison of RPA neutron scattering experiment. 58
4.4
Values of the scaling parameters {cn } required for quantitative comparison
of RPA neutron scattering calculations to experiment. . . . . . . . . . . . .
60
A.1 Crystal Field Parameters for Yb2 Ti2 O7 . . . . . . . . . . . . . . . . . . . . .
98
A.2 The wave-functions and energies of the crystal field states of Yb2 Ti2 O7 determined using the crystal field parameterization of Hodges et al. . . . . .
99
. .
58
A.3 The wave-functions and energies of the crystal field states of Yb2 Ti2 O7 determined using the crystal field parameterization of Cao et al. . . . . . . . 101
C.1 The character table of the Oh point group. . . . . . . . . . . . . . . . . . . 107
C.2 The characters of the symmetry operations of the Oh point group in the
representation Γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.3 The indirect Dzyaloshinskii-Moriya vectors for the pyrochlore lattice. . . . 116
D.1 Actions of the symmetry operations of the point group Oh on the elements
of the reducible representation Γ. . . . . . . . . . . . . . . . . . . . . . . . 119
ix
D.2 Actions of the symmetry operations of the point group Oh on the elements
of the reducible representation Γ continued. . . . . . . . . . . . . . . . . . 120
F.1 The hj0 (s)i magnetic form factor coefficients for Yb3+ . . . . . . . . . . . . 127
F.2 The hj2 (s)i magnetic form factor coefficients for Yb3+ . . . . . . . . . . . . 128
H.1
J
Jiso
, pd ,
JIsing JIsing
and
JDM
.
JIsing
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
x
List of Figures
1.1
Ising spins on a triangle with antiferromagnetic interactions. . . . . . . . .
3
1.2
The triangular lattice, the kagome lattice, and the pyrochlore lattice. These
are all lattices that have the capacity for geometric frustration due to the
presence of the triangular motif. . . . . . . . . . . . . . . . . . . . . . . . .
3
The interpenetrating pyrochlore lattices of the A and B ions of an A2 B2 O7
material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
The local Ising axes and easy planes of the four tetrahedral sublattice sites
of the pyrochlore lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.5
Specific heat (C/R) and inverse susceptibility (1/χ) of Yb2 Ti2 O7 . . . . . .
9
1.6
The Mössbauer absorption spectra of Yb2 Ti2 O7 at various temperatures. .
12
1.7
The fluctuation rate of the Yb3+ moment determined from Mössbauer absorption and µSR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
The measured values of the two independent components of the local susceptibility for Yb2 Ti2 O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Specific heat measurements performed on powder and single crystal samples
of Yb2 Ti2 O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.10 Quasi-elastic neutron scattering in the (h, k, k) plane at T = 1.4 K. . . . .
22
1.11 Temperature dependence of the rods of neutron scattering at selected points
in Q space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.3
1.4
1.8
1.9
1.12 The cubic unit cell of the pyrochlore lattice showing the alternating kagome
and triangular planes perpendicular to the [111 cubic body diagonal direction. 24
1.13 Diffuse neutron scattering intensity in the (h, h, l) plane. . . . . . . . . . .
xi
24
1.14 Inelastic neutron scattering along the [hhh] direction, for various strengths
of the magnetic field along [11̄0]. . . . . . . . . . . . . . . . . . . . . . . . .
25
1.15 Neutron Scattering in the (h, k, k) plane in zero external field at T = 1.4 K
and T = 9.1 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.1
An illustration of the nearest-neighbours of sublattice site a = 1 of the
pyrochlore lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
The quasi-elastic neutron scattering data at T = 1.4 K used to determine
the form of the magnetic interactions in Yb2 Ti2 O7 . . . . . . . . . . . . . .
52
S(Q) plotted for the sets of couplings, {Jn }, resulting from two different
simulated annealing runs. . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
The refined results of simulated annealing optimization for the CEF parameterizations of Hodges et al. and Cao et al. . . . . . . . . . . . . . . . . . .
57
Quasi-elastic neutron scattering computed at T = 1.4 K using selected results of simulated annealing fits to experiment. . . . . . . . . . . . . . . . .
59
Cuts through the (h, k, k) plane in reciprocal space comparing experimental
and RPA quasi-elastic neutron scattering. . . . . . . . . . . . . . . . . . .
61
The relationship between two-dimensional correlations and rods of scattering
intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.2
High symmetry directions of the pyrochlore lattice. . . . . . . . . . . . . .
68
5.3
Real space correlations functions S(r) and S⊥ (r), computed at T = 1.4 K,
plotted along three high symmetry directions of the pyrochlore lattice. . . .
69
RPA Quasi-elastic neutron scattering computed in three planes perpendicular to the direction [111] in reciprocal space. . . . . . . . . . . . . . . . .
71
Illustration of the splitting of the tetrahedral sublattice of the pyrochlore
lattice into α and β chains in the presence of a [110] magnetic field. . . . .
76
6.2
Calculations of the two components of χa with no interactions. . . . . . . .
78
6.3
Calculations of the two components of χa with magnetic interactions. . . .
80
6.4
Calculations of the RPA bulk susceptibility based on anisotropic and isotropic
exchange models compared to experiment. . . . . . . . . . . . . . . . . . .
81
4.1
4.2
4.3
4.4
4.5
5.1
5.4
6.1
xii
7.1
Monte Carlo simulation results. . . . . . . . . . . . . . . . . . . . . . . . .
84
7.2
The low temperature ground states of the classical anisotropic spin-1/2
model determined using Monte Carlo simulations and mean field calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Monte Carlo calculations of P (φ) for our bilinear exchange model with no
long-range magnetic dipolar interactions. . . . . . . . . . . . . . . . . . . .
86
Mean field neutron scattering and real space correlation function results for
the Heisenberg ferromagnet. . . . . . . . . . . . . . . . . . . . . . . . . . .
91
7.3
7.4
C.1 The indirect Dzyaloshinskii-Moriya vectors for the pyrochlore lattice. . . . 116
J
pd
DM
iso
, JIsing
, and JJIsing
as a function of x, the ratio of two
H.1 Plots of the ratios JJIsing
Slater-Koster parameters for the ranges x ≈ [−20, −0.5] and x ≈ [10, 30]. . 134
J
pd
iso
DM
H.2 Plot of the ratios JJIsing
, JIsing
, and JJIsing
Slater-Koster parameters for the range x ≈ [−0.5, 10]. . . . . . . . . . . . . 135
xiii
Chapter 1
Introduction - An Experimental
Overview of Yb2Ti2O7
Magnetic materials currently attract a great deal of interest in the field of condensed
matter theory as they provide an excellent sand box in which to explore the physics of
many bodied systems. Such many bodied systems display a wide variety of behaviours
that would not be seen if the constituent elements of the system were studied in isolation.
The most widely known collective magnetic phenomenon is simple ferromagnetism, where
the quantum mechanical magnetic moments of the ions in a material align to produce a
net magnetic moment. In the last decade considerable interest has been centred on one
particular group of magnetic materials, those which display frustration of the interactions
between the magnetic moments in the system. Frustrated materials are of great interest
due to the new and exotic states of matter they exhibit. These exotic states, such as the
spin ice state which is currently attracting a great deal of attention [1, 2, 3], are made
possible by the presence of frustration.
1.1
Frustration
Frustration is defined as the inability of a system of interacting elements to simultaneously
energetically minimize all of the interactions between the elements of the system. In
magnetic systems, frustration can have two sources. The first is competition between
different interactions present in the system and the second is the geometry of the system.
In this work we will focus on the second source, so called geometric frustration.
1
Geometric frustration arises in magnetic systems with fixed element geometries, such
as crystal lattices, where the geometry of the system prevents the simultaneous energetic
minimization of all of the interactions. This can lead to the presence of a highly degenerate
ground state manifold and the existence of strange magnetic phases. The classic example
of geometric frustration is the case of the global Ising antiferromagnet on a triangle. The
Hamiltonian of this system is given by
X
H=J
Si Sj ,
(1.1)
hi,ji
where J is the strength of the exchange interaction between the spins in the system, with
J < 0 being ferromagnetic exchange, and J > 0 being antiferromagnetic exchange. Si
and Sj are Ising spins of value ±1, at crystal lattice sites i, j, and the angle brackets in
the sum indicate that only nearest-neighbour pairs of spins are to be summed over and
double counting is to be avoided. It can be clearly seen that for a single pair of spins
i, j = 1, 2 with J > 0, the energy of the pair is minimized when S1 = +1 and S2 = −1.
If we consider three spins i, j = 1, 2, 3 on a triangle, see Fig. 1.1, we can see frustration
at work. If we start at site 1, setting S1 = +1 (an up arrow in Fig. 1.1), then proceeding
around the triangle, to satisfy the interaction between S1 and S2 , we must impose that
S2 = −1 (a down arrow in Fig. 1.1). Proceeding around the triangle, to satisfy the pairwise
interactions between S2 and S3 we must impose that S3 = +1. After this we are left only
with the pairwise interactions between S3 and S1 . If we then examine S3 and S1 , we can
see that they are both +1, so the interaction between them is not energetically minimized.
If we were to set either S1 or S3 to −1 in order to satisfy this interaction, one of the other
two pairwise interactions would not be in its minimum energy state. This means that the
system can never energetically minimize all of the pairwise interactions on the triangle,
thus it is frustrated.
In this case the frustration of the interactions leads to a ground state with six-fold
degeneracy, which can be seen by going through the process just described, starting at all
three sites on the triangle with both up and down spins. This is an example of a degenerate
manifold of ground states. If we were to extend the single triangle to a triangular lattice
(Fig. 1.2(a)), or some other lattice containing the triangular motif, such as the kagome
(Fig. 1.2(b)) or pyrochlore lattice (Fig. 1.2(c)), the degeneracy of the ground state grows
massively.
2
1
2
3
Figure 1.1: Ising spins on a triangle with antiferromagnetic interactions described by
Eqn. (1.1). Green bonds are those where the pairwise antiferromagnetic interaction between nearest-neighbours are energetically satisfied, the red bond shows the presence of a
pairwise interaction that is not in its energetic ground state. Up arrows are equivalent to
Si = +1, down arrows are equivalent to Si = −1, where i = 1, 2, 3.
(a)
(b)
(c)
Figure 1.2: The triangular lattice (a), the kagome lattice (b), and the pyrochlore lattice (c).
These are all lattices that have the capacity for geometric frustration due to the presence
of the triangular motif.
3
1.2
The Pyrochlore Lattice
The pyrochlore lattice is a crystal lattice that lends itself to geometrical frustration of
magnetic interactions because it is constructed of many triangular motifs assembled into a
lattice of corner sharing tetrahedra. For example, antiferromagnetic interactions between
both classical [4] and quantum [5] Heisenberg spins on the pyrochlore lattice have been
shown not to order down to the lowest temperatures.
The pyrochlore lattice is a non-Bravais lattice made up of corner sharing tetrahedra, as
shown in Fig. 1.2(c). It can be thought of as an FCC lattice with a four ion, tetrahedral
basis at each lattice site. The vectors that describe the FCC lattice sites Ri are defined
in Table 1.1, along with the vectors that describe the positions of the four sublattice sites
with respect to the FCC lattice sites, ra . Using this notation, we can index any site on
the pyrochlore lattice using the vector Rai = Ri + ra , and define the vector joining two
pyrochlore lattice sites as
Rab
(1.2)
ij = Ri + ra − Rj − rb
√
The nearest-neighbour distance for the pyrochlore lattice is 2rc /4, where rc = 10.026 Å is
the cubic unit cell dimension for Yb2 Ti2 O7 [6].
Table 1.1: FCC lattice vectors and tetrahedral sublattice vectors for the pyrochlore lattice.
rc = 10.026 Å is the cubic unit cell dimension [6].
FCC Lattice Vectors
R1 = r2c (0, 0, 0)
R2 = r2c (0, 1, 1)
R3 = r2c (1, 0, 1)
R4 = r2c (1, 1, 0)
Tetrahedral Sublattice Vectors
r1 = r4c (0, 0, 0)
r2 = r4c (0, 1, 1)
r3 = r4c (1, 0, 1)
r4 = r4c (1, 1, 0)
The reciprocal lattice vectors Gα for the FCC lattice are given in Table 1.2. This allows
us to break any reciprocal space vector Q into two components Q = q + G, where G is
a linear combination of vectors Gα , and q is a vector inside the first Brillouin zone of the
FCC lattice. The first Brillouin zone is a reciprocal space nearest-neighbour Wigner-Seitz
cell constructed about the origin, or any other reciprocal lattice point. The reciprocal
2π·R ×R
lattice vectors Gα are constructed using the formula Gk−1 = Ri ·(Rji ×Rjk ) , for i, j, k = 2, 3, 4.
4
Table 1.2: FCC reciprocal lattice vectors Gα . Once again, rc = 10.026 Å is the cubic unit
cell dimension [6].
FCC Reciprocal Lattice Vectors
G1 = 2π
(−1, 1, 1)
rc
(1, −1, 1)
G2 = 2π
rc
2π
G3 = rc (1, 1, −1)
1.3
Rare Earth Pyrochlores
One particular family of frustrated magnetic materials that has been the subject of a great
deal of interest is the rare earth pyrochlore oxides. These are materials with the chemical
formula A2 B2 O7 , where A is a rare earth ion (Ho,Dy,Tb,Gd, or Yb) or yttrium (Yt), and
B is a tetravalent transition metal ion (Ti, Sn, Mo, or Mn). These materials are described
by the F d3̄m space group [7], with the A ions occupying the 16c sites, and the B ions
occupying the 16d sites of this space group, forming interpenetrating pyrochlore lattices as
seen in Fig. 1.3.
Figure 1.3: The interpenetrating pyrochlore lattices of the A ions (blue) and B ions (red)
of an A2 B2 O7 material [8].
5
Members of the rare earth pyrochlore oxide family of materials have been found to exhibit exotic phenomena at low temperature such as spin liquid behaviour in Tb2 Ti2 O7 [9],
and other exotic phenomena that arise due to material specific effects [10]. These include
spin glass behaviour in Y2 Mo2 O7 [11], spin ice in Ho2 Ti2 O7 , Dy2 Ti2 O7 , Dy2 Ti2 O7 ,and
Ho2 Sn2 O7 [1, 10], and long-range order (LRO) with persistent low-temperature spin dynamics in Gd2 Sn2 O7 and Er2 Ti2 O7 [12, 13].
1.4
Ytterbium Titanate - Yb2Ti2O7
Ytterbium Titanate (Yb2 Ti2 O7 ) is another member of the rare earth pyrochlore family
of materials that displays exotic behaviour at low temperatures. In this material various
experimental measurements have found evidence of a magnetic phase transition at low
temperatures [14, 15, 16, 17]. Interestingly, no evidence of magnetic LRO [6] has been
found below this transition. We will discuss these experiment in greater detail later in this
study, but first we must establish some of the basic properties of Yb2 Ti2 O7 .
In Yb2 Ti2 O7 the Yb3+ ions are the only magnetic ions, and as previously discussed, they
reside on a pyrochlore lattice of corner sharing tetrahedra. The electronic configuration of
the Yb3+ ions in this material is 4f 13 , 2 F7/2 . This state has a high contribution from orbital
angular momentum, so the good quantumpnumber is J = 7/2. The paramagnetic moment
of the Yb3+ ions is then given by µ = gJ J (J + 1)µB ≈ 4.54µB . This is consistent with
experiments which find that the effective magnetic moment of the Yb3+ ions in ∼ 4µB at
room temperatures, decreasing to ∼ 3µB at liquid helium temperatures [18]. The ground
state of the crystal field (CEF) of Yb2 Ti2 O7 is a Kramers doublet [14]. This doublet
structure is imposed by the fact that the Yb3+ ion has an odd number of electrons in the
valence shell. The findings of various experimental studies on the CEF of Yb2 Ti2 O7 are
that the ground state Kramers doublet of the Yb3+ ions is locally planar [19, 20], meaning
that the magnetic moments prefer to lie in planes perpendicular to the local [111] directions
(defined in Table 1.3) at each corner of the tetrahedral sublattice, as shown in Fig. 1.4. If
we treat the ground state doublet as an effective spin-1/2 state, this planar nature of the
ground state doublet can be described by a g tensor, with two components gk and g⊥ that
describe the preference of the effective spin to lie along the local z axis (gk ), or in the local
x − y plane (g⊥ ). The g tensor and other properties of the CEF experienced by these ions
are discussed in Appendix A.
6
Figure 1.4: The local Ising axes (red) and easy planes (blue) of the four tetrahedral sublattice sites of the pyrochlore lattice [21].
Table 1.3: Local coordinates for the four sublattice sites of the pyrochlore lattice.
Sublattice
a=1
a=2
x̂a
√1 (−1, −1, 2)
6
√1 (1, −1, 0)
2
√1 (1, 1, 1)
3
√1 (1, 1, 2)
6
√1 (−1, 1, 0)
2
√1 (−1, −1, 1)
3
ŷa
ẑa
7
a=3
√1
6
√1
2
√1
3
(1, −1, −2)
(−1, −1, 0)
(−1, 1, −1)
a=4
√1 (−1, 1, −2)
6
√1 (1, 1, 0)
2
√1 (1, −1, −1)
3
1.5
Previous Work on Yb2Ti2O7
The first published experiments on Yb2 Ti2 O7 were magnetic susceptibility measurements
collected between T = 2 K and T = 1400 K on a powdered sample of Yb2 Ti2 O7 , published in 1968 by Townsend et al. [22]. These measurements were used to attempt a
parameterization of the CEF experienced by the Yb3+ ion. This work admitted that the
parameterization produced from the analysis of these measurements should be treated
with caution. Recent (at the time) measurements on powdered samples of the material
ytterbium gallium garnet showed similar results in terms of the crystal field structure to
that of Yb2 Ti2 O7 determined in this work, and the ytterbium gallium garnet results were
contradicted by spectroscopic measurements on said material [22].
The next published measurements of interest to this work were specific heat measurements performed by Bløte et al. in 1969 [16], these measurements are shown in Fig. 1.5(a).
The most interesting feature of these measurements is the peak in the specific heat at
T ∼ 214 mK [16], indicating the presence of a first order phase transition. In addition to
specific heat (Cv ) measurements, Ref. [16] also contains measurements of the bulk susceptibility χ as shown in Fig. 1.5(b). From these measurements the Curie-Weiss temperature
θCW is determined to be θCW = 0.4 ± 0.1 K. θCW is defined as the temperature axis intercept of a linear fit to the inverse susceptibility and is an indication of strength and type of
magnetic interactions in a material. A positive θCW indicates that the magnetic interactions in a material are mainly ferromagnetic. From Cv , values of the magnetic energy gain
|∆E/R|, and the Heisenberg exchange energy |J/k| were computed, obtaining values of
|∆E/R| = 1.8 K and |J/k| = 1.3 ± 0.1 K. It is noted that these are much larger than the
transition temperature. J/k can also be computed from the susceptibility data, yielding
J/k = 0.13 K. The fact that this is so different from the value extracted from Cv is taken
as evidence that the exchange interaction in this material may be highly anisotropic. A
2
constraint on the g tensor is also provided, gk2 + 2g⊥
= 17 [16]. The contradictions between
the observed temperature of the phase transition and estimated values of the exchange
energy and magnetic energy provide the first hint that something interesting may be going
on in this material beyond a simple first order phase transition to a state with long-range
order (LRO), and serve as the starting point for this work.
After these measurements, 170 Yb Mössbauer absorption measurements were performed
by Dunlap et al. in 1978 [23], in an effort to better understand the CEF experienced by
the Yb3+ ions in Yb2 Ti2 O7 . The analysis of these measurements concluded that the CEF
states of the Yb3+ ion consist of two low energy doublets, separated by a gap of ∆E ∼ 20
K, with higher energy states lying at ∆E ∼ 130 K, and ∆E ∼ 750 K [23]. This analysis
used four parameters to describe the the CEF experienced by the Yb3+ ions, B20 ,B40 ,B60 ,
8
(a)
(b)
Figure 1.5: Figures reproduced from Ref. [16]. (a) is a plot of the specific heat (C/R), and
(b) is a plot of the inverse susceptibility (1/χ) of Yb2 Ti2 O7 in arbitrary units. The dashed
line in (a) on the right side below 0.2 K represents the hyperfine splitting contribution to Cv
and the remaining heat capacity after the removal of the hyperfine splitting contribution.
On the left side of the graph the dashed line represents an extrapolation of the behaviour,
as does the dash-dotted line. The two extrapolations correspond to different values of
the entropy, with integration along the dashed line yielding S = 0.671R, and integration
along the dash-dotted line yielding S = R ln 2. The solid line in (b) represents the high
temperature fit between T = 2 K and T = 3.5 K, used to determine θCW = 0.4 ± 0.1 K.
and B66 , which are used to define a linear combination of Stevens-operator equivalents [24]
to describe the crystal field [23]. The work of Sengupta, published in 1999 [25], explains
that four Stevens-operator equivalents are insufficient to describe the CEF of Yb2 Ti2 O7 ,
given the D3d symmetry of the oxygen ions surrounding the Yb3+ ion [25]. Ref. [25] goes
on to explain that a CEF parameterization based on six Stevens-operator equivalents is
required, and attempts to extract the coefficients of these terms, A02 , A04 , A34 , A06 , A36 , and
A66 , from the magnetic susceptibility data of Ref. [22], and the Mössbauer absorption data
of Ref. [23]. The conclusions of these works with respect to the form of the CEF in Yb2 Ti2 O7 have subsequently been shown to be incorrect by more recent work [19, 20, 26], which
will be discussed later.
The next work on Yb2 Ti2 O7 , is that of Siddharthan et al., also published in 1999 [27],
9
who performed magnetic susceptibility and specific heat measurements on Yb2 Ti2 O7 and
other rare earth pyrochlores. They then used Monte Carlo simulations based on an Ising
model with super-exchange and long-range magnetic dipolar interactions to attempt to
explain the observed behaviours of the specific heat of Yb2 Ti2 O7 . This was done despite
citing evidence from CEF calculations [28] that the ground state doublet of the Yb3+ CEF
should be local XY planar in nature, not local Ising-like [27]. For the case of Yb2 Ti2 O7 , they assume only long-range dipolar interactions are at play, and obtain reasonable
agreement with experiment on the position of the peak in the specific heat [27]. Their final
comment on Yb2 Ti2 O7 is that further calculations are in progress, but if they were ever
published, we have not been able to locate them.
After the work by Siddharthan [27], Bramwell et al. [18] published in 2000 measurements of the magnetic susceptibility and bulk magnetization, M , of several rare earth
pyrochlores including Yb2 Ti2 O7 . From these measurements, they computed various quantities, including the free ion moment at various temperatures, and θCW = 0.7 K. An attempt
was made to fit the observed field dependence of the magnetization at various temperatures
to a model of an effective spin-1/2 doublet with an effective g value (the diagonal elements
of the g tensor are assumed to be identical). Performing this fit is shown to result in a
different value of g at each temperature, with an optimal value over all the temperatures
of g = 7.2 [18]. This variation of g with temperature is taken to indicate a failure in the
model and as an indication that g⊥ may be significantly different from gk in this material.
The next set of publications on Yb2 Ti2 O7 were research by Hodges et al., Refs. [14, 15,
20]. These works present a very comprehensive set of measurements on Yb2 Ti2 O7 , including 170 Yb mössbauer absorption, magnetic susceptibility, magnetization, 172 Yb perturbed
angular correlation (PAC), and neutron scattering measurements.
The first work, Ref. [20] published in 2001, presents 170 Yb Mössbauer absorption, 172 Yb
perturbed angular correlation (PAC) measurements, magnetic susceptibility, and specific
heat measurements performed on polycrystalline samples of (Y0.99 Yb0.01 )Ti2 O7 and Yb2 Ti2 O7 in an effort to quantify the CEF structure and magnetic interaction strength in
Yb2 Ti2 O7 . The first measurements presented in this work are 170 Yb Mössbauer absorption measurements on (Y0.99 Yb0.01 )Ti2 O7 , which are used to determine the g tensor for
isolated Yb3+ moments, finding gk = 1.79, g⊥ = 4.27. The next experiments presented
are bulk DC susceptibility and magnetization measurements in Yb2 Ti2 O7 . The susceptibility was measured from T = 2.5 K to T = 10 K, finding θCW = 0.75 K, indicating
that the interactions between the Yb3+ moments are ferromagnetic in nature. Fits to the
magnetization at several temperatures, using the previously determined planar g tensor,
show that adding ferromagnetic interactions between the Yb3+ moments fits the data quite
well. This provides further evidence for net ferromagnetic interactions in this material. A
10
suggestion that these interactions should be anisotropic in nature is made, but introducing
this anisotropy does not improve the fit, so the authors abandon the idea. The interaction
~ → H
~ + λ~µ, with λ = 0.31(4) T/µB ,
term used to perform these fits is expressed as H
so that it encompasses both exchange and long-range dipolar interactions. In addition to
measurements on polycrystalline samples, magnetization measurements on a single crystal
of Yb2 Ti2 O7 are discussed, with the conclusion that they do not yield any new information. Finally 172 Yb PAC measurements are reported. These measurements are used to
quantify the thermal dependence of the quadrupole hyperfine interaction of the 172 Yb nucleus, which was found by the Mössbauer absorption measurements to be quite small. In
addition, the PAC measurements are fitted using six free parameters, B20 , B40 , B43 , B60 , B63 ,
and B66 , corresponding to the inclusion of six different Stevens-operator equivalent in the
CEF parameterization, as required by the D3d symmetry of the Yb3+ ion site. Starting
from an unpublished set of CEF parameters in Ref. [29], the authors obtain a good fit to
the PAC measurements, obtaining a CEF parameterization that is discussed in detail in
Appendix A. This CEF parameterization yields gk = 1.77, g⊥ = 4.18 for the ground state
doublet, with a gap between the ground state and first excited state of ∆E = 620 K. The
authors conclude by stating that previously published CEF parameterizations [22, 23, 25]
are not compatible with the experimental observations presented in this work and this is
the reason we do not consider the work of Refs. [22, 23, 25] any further in this study.
The second work in this set of publications, Ref. [14] also published in 2001, presents
Yb Mössbauer absorption and 172 Yb PAC measurements. Mössbauer absorption spectroscopy data for both (Y0.99 Yb0.01 )Ti2 O7 and Yb2 Ti2 O7 is presented, with the work on
(Y0.99 Yb0.01 )Ti2 O7 being the same as that in Ref. [20]. The Mössbauer absorption spectra
for Yb2 Ti2 O7 are new to this work, previous works reported only Mössbauer absorption
spectra for (Y0.99 Yb0.01 )Ti2 O7 , with absorption spectra for polycrystalline samples of Yb2 Ti2 O7 at various temperatures from T = 4.2 K down to T = 0.036 K being reported,
as shown in Fig. 1.6. A marked change in the Mössbauer absorption spectra is reported
at T ∼ 0.3 K. Above this temperature, the spectrum is characterized by a single narrow
line, whose width increases as the temperature decreases, while below T ∼ 0.26 K a five
line hyperfine-field sub-spectrum appears, completely replacing the single line spectrum
by T ∼ 0.2 K. The broadening of the single line spectrum for T > 0.3 K is attributed to
fluctuations of the Yb3+ moments and the fluctuation rate is extracted from the width of
these lines. The five line spectrum observed for T < 0.2 K is indicative of a hyperfinefield that is static on the time scale of Mössbauer spectroscopy so the fluctuation rate
of the Yb3+ moments cannot be measured below this temperature. The strength of the
hyperfine-field leading to this five line spectrum is extracted, yielding a Yb3+ moment size
in the low temperature phase of 1.15µB . The temperature dependence of the size of the
170
11
moment is determined, finding that the moment goes from zero to ∼ 1.15µB very sharply
at T = 0.26 K, consistent with the observed peak in the specific heat in Ref. [16]. Finally,
using the size of the ground state moment and the g tensor determined in Ref. [20] the
direction of the moment with respect to the local [111] z axis is determined via the equation
h
i1/2
M = 0.5 gk2 cos2 α + 2g⊥ sin2 α
, where α, the angle between the moment and the local
◦
◦
z axis, is found to be 22 ± 3 .
Figure 1.6: Figure reproduced from Ref. [14]. The Mössbauer absorption spectra of Yb2 Ti2 O7 at various temperatures. The solid lines show the combination of single line and five
line spectra that make up the data at T = 0.30 K.
The third work of this set, Ref. [15] published in 2002, presents the previously discussed
Yb Mössbauer absorption, along with new neutron scattering, and muon spin resonance
(µSR) measurements, in an effort to better understand the nature of the phase transition
and low temperature phase of Yb2 Ti2 O7 . The neutron scattering measurements presented
were performed on a powder sample of Yb2 Ti2 O7 . Neutron scattering measures the correlation between magnetic moments in materials, as such it is admirably suited to detecting
the presence of magnetic correlations or LRO in materials. If magnetic LRO exists in a
material, it is typically evidenced by magnetic Bragg peaks in the elastic neutron scattering
cross section, the component of the neutron scattering cross section where the energy of
the neutrons is conserved. These peaks are either superimposed on nuclear Bragg peaks,
which arise from the nuclear magnetic moments in the material, or at new positions in
170
12
reciprocal space. The nuclear Bragg peaks are indicative of the crystalline order of the
material. Analysis of the positions of these magnetic Bragg peaks can reveal the type of
magnetic order that is present in the material. The neutron scattering measurements of
Ref. [15] reveal that the crystal structure of Yb2 Ti2 O7 does not change with temperature,
and that no magnetic Bragg peaks are found to appear below T = 0.2 K, indicating that
no magnetic LRO is present in the low temperature phase. This means that the static
hyperfine field observed by Mössbauer absorption in Ref. [14] is not due to LRO, but to
slowing of the fluctuation rate of the Yb3+ moments to level a below that detectable via
Mössbauer absorption. In addition to this neutron scattering data, µSR measurements are
presented and used to determine the fluctuation rate of the Yb3+ moments below T = 0.2
K in the range where Mössbauer absorption is incapable of distinguishing the fluctuation
rate. These µSR measurements find a non-zero, temperature independent Yb3+ moment
fluctuation rate of ∼ 1000 MHz, as shown in Fig. 1.7. This temperature independent fluctuation rate below T = 0.24 K is a further indication that something very strange is going
on in the low temperature phase of Yb2 Ti2 O7 . Ref. [17] published in 2006 contains additional work on µSR data in various magnetic fields at T ≈ 0.07 K. This work was done to
determine if the persistent spin dynamics observed in Ref. [15] were due to a large density
of states at low energies. The work finds that a gaussian-broadened gaussian (GBG) model,
rather than the traditional Kubo-Toyabe model associated with an ensemble of slowly fluctuating moments, fits the experimental data quite well below the temperature of the peak
in the specific heat reported in Ref. [16]. The GBG model assumes that each muon sees
a different environment, described by a dynamic Kubo-Toyabe function, and that fields
in these environments are characterized by a gaussian distribution. The most interesting
finding of this work is that the GBG model only works if a magnetic field strength less
than that of the actual applied field is used in the model, but no interpretation of this
effect is provided.
In 2003 Bonville et al. published a review article, Ref. [30], discussing experiments on
both Yb2 Ti2 O7 and Gd2 Ti2 O7 . This work provides an excellent summary and discussion
of the results of most of the previous work on Yb2 Ti2 O7 . This review does contain some
new work, such as zero-field and field-cooled measurements of the bulk DC susceptibility,
showing irreversibilities similar to those shown in glassy materials [30], and neutron scattering measurements on a single crystal of Yb2 Ti2 O7 , which find no evidence of Bragg peaks
down the lowest temperatures measured (T = 0.04 K). The neutron scattering measurements include quasi-elastic neutron scattering data in the paramagnetic phase of Yb2 Ti2 O7
(T > 0.3 K). Quasi-elastic neutron scattering is neutron scattering where the difference in
energy of the of the incident and scattered neutrons is very small. This quantity is what
is detected in many elastic scattering experiments due to the finite energy resolution of
13
Figure 1.7: Figure reproduced from Ref. [15]. The fluctuation rate of the Yb3+ moment
determined from Mössbauer absorption, νM , and from µSR, νµ . A sudden change is seen
to take place at T ∼ 0.24 K, consistent with the peak in the specific heat observed in
Ref. [16]. Below T ∼ 0.24 K, the fluctuation rate falls below the minimum value detectable
by Mössbauer absorption, indicated by the dashed line.
the detectors. The single crystal paramagnetic neutron scattering presented in this work
is the same data, but at much lower resolution, that is presented in our published work,
Ref. [31], and which forms a key part of this study. The most interesting feature of this
neutron scattering data, shown in Fig. 1.10, is the [111] rod of scattering intensity, seen
to develop at T = 25 K, which increases in intensity as T decreases, as seen in Fig. 1.11.
The rod-like feature is taken as evidence of bi-dimensional real space correlations between
Yb3+ moments, an idea that will be tested in this study. Reference [32] is essentially the
same document, but with significantly less neutron scattering data being presented. This
data is discussed further in the next section.
Also in 2003, Yasui et al. published neutron scattering, magnetization, and susceptibility measurements on a single crystal of Yb2 Ti2 O7 in Ref. [33]. This somewhat controversial
work claims to detect ferromagnetic order in the low temperature phase of Yb2 Ti2 O7 via
the existence of magnetic Bragg peaks in low temperature neutron scattering data. Bulk
DC susceptibility measurements presented in this work yield θCW = 0.59 K, and the real
part of the AC susceptibility shows an anomaly at T = 0.24 K that, along with magnetization measurements, is taken as evidence for a ferromagnetic phase transition at T = 0.24
K. The magnetization is also used to determine the strength of the inter-ionic exchange
in Yb2 Ti2 O7 in combination with the values of gk and g⊥ for the ground state doublet
14
of the CEF. A value for the interaction term λ is found, λ = 0.64 ± 0.1 µB /T, which is
approximately double the value found in Ref. [20], and the components of the g tensor
are found to be gk = 2.6 ± 0.4 and g⊥ = 3.9 ± 0.2. This is a much less anisotropic form
of the g tensor than was found in Ref. [20]. The neutron scattering measurements below
T = 0.03 K show no Bragg peaks other than nuclear Bragg peaks, but analysis of the
temperature dependence of the intensity of these peaks reveals magnetic intensity at several of the nuclear Bragg peaks. The distribution of the magnetic intensity at the nuclear
Bragg peaks is consistent with q = 0 ferromagnetic order. It is interesting to note that
the peaks were detected after letting the sample equilibrate at T = 0.03 K for 10h, a very
long time scale seemingly not taken into account in other experiments involving the low
temperature phase of Yb2 Ti2 O7 . This long time scale of the magnetic neutron scattering
intensity leads to hysteresis in the magnetic scattering intensities on cooling and warming.
Finally, typical time scales for changes of the neutron scattering intensity are reported
for two temperatures, T = 0.03 K and T = 1 K, where time scales of 120 min and 30
min are found respectively. This is interpreted as evidence that the motion of the Yb3+
moments slows as temperature decreases [33]. No information on how these time scales
were determined in given.
The work of Gardner et al., Ref. [6] published in 2004, presents neutron spin echo
(NSE) and polarized neutron scattering measurements designed to refute the findings of
Ref. [33]. NSE measures S (Q, t), which contains information about the temporal and
spatial correlations between magnetic moments. The presented NSE measurements reveal
that S (Q) increases with decreasing |Q|, consistent with ferromagnetic interactions, as
found in Ref. [20]. The presented NSE measurements also reveal signal relaxation times
of less than 4 ps at both T = 180 mK and T = 1 K, inconsistent with the presence of
ferromagnetic LRO in the low temperature phase. Polarization analysis of the neutron
beam is also inconsistent with ferromagnetic LRO in the low temperature phase, as no loss
of polarization is found, which would occur in the case of a polycrystalline ferromagnet.
Polarized neutron scattering does find evidence of a magnetic contribution to the Bragg
scattering at q = (1, 1, 1) at T = 90 mK, consistent with Ref. [33], but not at any other
Bragg peaks. In order to explain all of these findings, two scenarios are proposed: that the
system consists of small domains of ordered (but not ferromagnetically ordered) moments
surrounded by fluctuating moments, or that the observed magnetic contribution to the
Bragg scattering is due to nuclear ordering, and not electronic moment ordering1 .
1
Important: When taken in combination, the results of Ref. [6] and Ref. [33] present a very muddy
picture of whether there in fact exists any magnetic LRO in the low temperature phase of Yb2 Ti2 O7 .
Ref. [33] appears to present clear evidence of ferromagnetic LRO in the low temperature phase of Yb2 Ti2 O7 . Ref. [6] on the other hand reports quite clear evidence, in the form of neutron spin echo (NSE) and
15
In 2004, Malkin et al. published an article on Yb2 Ti2 O7 reporting the use of optical
spectroscopy to determine the CEF energy levels of Yb2 Ti2 O7 , Ref. [26]. This study was
inspired by marked differences between the CEF structure of Yb2 Ti2 O7 found in Ref. [20],
and the structure that would be expected based on those of other rare earth pyrochlore
materials. These differences cannot be explained on the basis of the effect of changes in
lattice parameters between compounds on the positions of the other ions in the material
surrounding the rare earth ions. IR absorption spectroscopy was performed on Yb2 Ti2 O7
and IR fluorescence spectroscopy was performed on Y2 Ti2 O7 (1% Yb). Beginning from a
CEF parameterization of Ho2 Ti2 O7 , a set of CEF parameters, B20 , B40 , B43 , B60 , B63 , and
B66 , that respect the D3d symmetry of the Yb3+ site are refined to fit the fluorescence spectroscopy data. The ground state doublet of the refined CEF parameterization is described
by gk = 1.836 and g⊥ = 4.282, similar to the g tensor parameters found in Ref. [20], but the
parameters B20 , B40 , B43 , B60 , B63 , and B66 are quite different from those found in Ref. [20].
The differences between the CEF parameterization of this work and that of Ref. [20] provide the impetus for the second CEF parameterization of Yb2 Ti2 O7 used in this study.
This second CEF parameterization, discussed in Appendix A is refined from 170 Yb PAC
measurements and the CEF parameterization of Ho2 Ti2 O7 in Ref. [34].
The next set of publications on Yb2 Ti2 O7 are a pair of publications on local susceptibility (χa ) measurements by Cao et al., Refs. [19, 34]. χa is defined as the coupling between
a single magnetic moment at a sublattice site a and the external field H, by the equation
Ma = χa H, where a = 1, 2, 3, 4 is the tetrahedral sublattice label. It is extracted from
polarized neutron scattering measurements via the method of Ref. [35], and is explained
further in Chapter 6.1.
The first work of Cao et al., Ref. [19] published in 2009, reports χa measurements
on several rare earth pyrochlore oxide materials including Yb2 Ti2 O7 . The reported measurements of χa for Yb2 Ti2 O7 are shown in Fig. 1.8. This work presents a new CEF
parameterization for Yb2 Ti2 O7 that respects the D3d symmetry of the Yb3+ lattice site.
This parameterization is similar to that of Ref. [26], and is determined in a similar manner,
starting from a CEF parameterization of Ho2 Ti2 O7 and adjusting the CEF parameters for
that material to fit PAC measurements reported in Ref. [20]. The details of this CEF parameterization are discussed in detail in Appendix A. The ground state doublet of this CEF
parametrization is described by a g tensor that is planar in nature, as found in other CEF
beam polarization measurements, that no ferromagnetic LRO is present in the low temperature phase of
Yb2 Ti2 O7 . While Ref. [6] does not appear to collect data on the same time scales as Ref. [33], and does
find some evidence of magnetic ordering of an unknown type, we consider the NSE and beam polarization
results of Ref. [6] to be a sufficient contradiction of the results of Ref. [33] to conclude that no clear
experimental evidence of magnetic LRO in the low temperature of Yb2 Ti2 O7 has been found.
16
parameterizations [20], with gk = 2.24 and gk = 3.98. If this CEF parameterization is used
to compute the local susceptibility via mean field theory, it can be seen in Fig. 1.8 that the
fit fails at low temperatures. The authors attempt to eliminate this disagreement between
experiment by adding a term to the magnetic field, similar to the method of Refs. [20, 33],
←
→
but incorporating anisotropic exchange, to yield Heff
i = H+ λ ·mi , where H is the applied
←
→
magnetic field, mi is the magnetic moment at lattice site i, and λ is a 3 × 3 tensor which
describes the exchange field at that lattice site. In the local sublattice coordinate system,
←
→
λ has two components λk and λ⊥ . It is found that λk = 2.5 T/µB and λ⊥ = −0.05 T/µB
yields a good fit to the experimental data, as seen in Fig. 1.8. This is claimed by the
authors to be the first concrete evidence of anisotropic exchange interactions in Yb2 Ti2 O7 .
However, it is important to note that this model of the interactions does not properly take
into account the tetrahedral sublattice structure of the pyrochlore lattice. This sublattice
structure is very important to the physics of this material, due to the different CEF easy
planes at each corner of the tetrahedral sublattice, so any model that does not include this
structure does not correctly describe Yb2 Ti2 O7 .
Figure 1.8: Figure reproduced from Ref. [19]. The measured values of the two independent
components of χa for Yb2 Ti2 O7 , χ⊥ and χk (symbols), along with fits to χ⊥ and χk based
on models including only the CEF parameterization presented in Ref. [19] (dashed lines),
and a form of anisotropic exchange presented in Ref. [19] (solid lines). The insets are
magnetization ellipsoids, whose shape is determined by the form of χa , at 5 K (bottom)
and 250 K (top). The arrows represent the field induced moment directions.
The second work of Cao et al., Ref. [34] also published in 2009, concentrates on Yb2 Ti2 O7 , and reports the χa measurements of Ref. [19] along with measurements of the DC bulk
17
susceptibility of powder samples of Yb2 Ti2 O7 . The main component of this work is the
computation of the powder susceptibility from the anisotropic exchange model proposed
in Ref. [19]. They find that this model fits the experimental data better than either
ferromagnetic or antiferromagnetic Heisenberg exchange, but they restrict themselves to
Heisenberg models of unit strength (i.e. λk = λ⊥ = ±1 T/µB ) making the failure of these
models somewhat artificial, as the strength of the exchange is not allowed to vary to fit the
data. Based on the fact that the local Ising term of the anisotropic exchange model they
utilize (λk ) is so much larger that the local planar term (λ⊥ ), they propose that Yb2 Ti2 O7
may be something called an ‘exchange spin ice’. Such a material would be one where the
interactions lead to local Ising behaviour, rather than the CEF leading to this behaviour
as occurs in canonical spin ice materials [36]. It is argued that, given the magnitude of
the magnetic moments in the low temperature phase of Yb2 Ti2 O7 found in other works
[20], the different anisotropy of the ground state g tensor found in Ref. [19] allows for
the Yb3+ moments to lie along the h111i axis at each corner of a tetrahedron, as occurs
in the spin ice materials. Finally, a discussion of the implications of this model for the
low temperature spin dynamics of Yb2 Ti2 O7 is presented. It is argued that the large easy
plane terms of the g tensor will allow rapid fluctuations between spin up and spin down
local Ising states that cannot occur in conventional spin ices, perhaps explaining the low
temperature fluctuations observed in µSR experiments [15]. As we discussed earlier, the
exchange model of this work is based on does not respect the sublattice structure of the
pyrochlore lattice, so any discussion relying on the importance of the Ising component of
the anisotropic exchange model they present should be considered suspect because it does
not account for the presence of multiple sublattices in the pyrochlore lattice.
The work of Malkin et al., Ref. [37] published in 2010, takes the local susceptibility
measurements of Ref. [19] for several of the rare earth pyrochlore oxide materials, and
performs an improved fit to the data based on an anisotropic exchange model that does
account for the sublattice structure of the pyrochlore lattice. The form of the anisotropic
exchange used in this model is also constructed to obey the symmetry of the space group
F d3̄m, which describes the pyrochlore lattice. This symmetry consideration is incomplete
though, as the authors impose an artificial symmetry on two of the four symmetry allowed
bilinear exchange terms. They also fail to include a fourth symmetry allowed interaction
all together. In order to understand this imposed symmetry, and why it is artificial, we
observe that Ref. [37] defines three different exchange couplings λ⊥,1 , λ⊥,2 , and λk . If we
relate these to the symmetry allowed nearest-neighbour bilinear exchange interactions on
18
the pyrochlore lattice defined in Appendix C we find that (Eqns. A.1-A.3 in Ref. [38])
1
λ⊥,1 = JIsing + Jiso + Jpd ,
3
1
λ⊥,2 = Jiso + Jpd ,
2
2
λk = − JIsing + Jiso − 2Jpd ,
3
(1.3)
(1.4)
(1.5)
where JIsing , Jiso , and Jpd are the strengths of three different symmetry allowed exchange
terms defined in Appendix C. Reference [37] imposes the constraint λ⊥,1 = λ⊥,2 , which
artificially breaks the symmetry of the pyrochlore lattice and reduces the number of free
coupling variables by one. For the case of Yb2 Ti2 O7 , the authors state that they have
determined exchange couplings that are much larger than those reported for the other rare
earth pyrochlore oxides, but they do not report the actual values of these couplings.
Finally we come to specific heat measurements on powder and single crystal samples
of Yb2 Ti2 O7 presented in the Ph.D. thesis of J. Quilliam, Ref. [39] published in 2010,
and reproduced in Fig. 1.9. The most interesting feature of this data is that instead of
the sharp peak in Cv observed in both sets of powder data, the Cv measurements on a
single crystal of Yb2 Ti2 O7 show a broad peak around T = 180 mK. The single crystal
used in these measurements is part of the same crystal used to perform neutron scattering
measurements by Ross et al. in Ref. [40]. These measurements find a change in the form
of the diffuse neutron scattering, from rods indicative of 2-dimension correlations, to 3dimension correlations, but not long-range order, at a temperature consistent with the
feature observed in the single crystal specific heat data. The powder sample specific heat
reported in this work was measured using the powdered Yb2 Ti2 O7 used to produce the
single crystal, and shows a sharp peak at T = 270 mK. This marked difference between
the powder and single crystal measurements is quite interesting, and in combination with
the variations in the temperature at which the phase transition occurs between different
samples of Yb2 Ti2 O7 [16, 33] perhaps indicates issues with the quality of Yb2 Ti2 O7 powder
and single crystal samples.
1.6
Paramagnetic Neutron Scattering
We now come to the experiments of greatest interest to this study, quasi-elastic neutron
scattering measurements performed on single crystals of Yb2 Ti2 O7 at temperatures above
that of the phase transition observed in the specific heat.
19
Figure 1.9: Figure reproduced from Ref. [39]. Specific heat measurements performed
on powder and single crystal samples of Yb2 Ti2 O7 . The specific heat measurements of
Ref. [16], shown in Fig. 1.5, are also included for comparison. The figure has been modified to clarify the labels of the various measurements presented.
As briefly discussed earlier on pg. 13, neutron scattering measures correlations between
magnetic moments. More specifically, it measures the correlations between the components
of magnetic moments perpendicular to the scattering wave-vector Q. Neutron scattering
experiments are performed by placing a material sample into a beam of neutrons, either
from a nuclear reactor, or from a spallation source. These neutrons posses magnetic moments (spins), µn , that interact with the magnetic fields in the sample and are scattered.
These scattered neutrons are then detected in ways that allow the momentum and energy
of the scattered neutrons to be determined, such as triple axis diffractometers as in Ref. [31]
or time-of-flight detectors as in Ref. [40].
The neutron scattering experiments considered in this work are quasi-elastic neutron
scattering, neutron scattering with very small but not completely zero energy transfer
from the neutron to the sample. The quantity measured in experiment is the differential
d2 σ
neutron scattering cross section dΩdE
. In terms of experiment, this quantity is the number
of neutrons counted over a solid angle range dΩ and energy range dE, divided by the
20
incident neutron flux, or [41]
d2 σ
dΩdE
=
δN
Wksn ,k0 s0n .
|j (k, sn ) |dΩdE
(1.6)
δN is the number of neutrons incident on a detector in the ranges dΩ and dE, j (ksn ) is
the incident flux of neutrons with energy ~k and spin sn , and Wksn ,k0 s0n is the probability of
the neutron undergoing a transition from the state |ksn i to the state |k0 s0n i, which describe
the incident and scattered spin and momentum states of the neutrons respectively. The
microscopic physics behind this formulation of the neutron scattering cross section will be
covered in the next chapter.
Quasi-elastic neutron scattering measurements performed on single crystals of Yb2 Ti2 O7 were first presented in Ref. [30], as discussed in in the previous section. These
measurements were performed on a single crystal sample at the Institute Laue Langevin
(ILL), in Grenoble, France. These measurements reveal a very interesting feature; a rod of
scattering in the (h, k, k) plane along the [111] direction. This rod is found to develop at
T ≈ 25 K, and increases in intensity as the temperature decreases, even though its width
does not change significantly [30]. It is also stated that the intensity of the rods peaks at
T = 2 − 3 K, the same temperature as a broad feature in the specific heat observed in
Ref. [16]. The scattering in the (h, k, k) plane reported in Ref. [30] is shown in Fig. 1.10.
The temperature dependence of the intensity of the rods is reported in Ref. [30] for two
points in Q space, Q = (1.5, 1.5, 1.5) and Q = (1.9, 1.9, 1.9), and these results are shown
in Fig. 1.11.
The next reported set of neutron scattering measurements on the paramagnetic phase of
Yb2 Ti2 O7 is the work of Ross et al., Ref. [40] published in 2009. In this work, many different
neutron scattering measurements are reported, not only quasi-elastic neutron scattering,
but also inelastic neutron scattering for temperatures from 4 K, all the way down to 30
mK. These measurements were performed both in zero magnetic field, and in an external
magnetic field applied along the [110] direction. These measurements were performed
using the disk chopper spectrometer at the NIST Center for Neutron Research on a single
crystal of Yb2 Ti2 O7 grown at McMaster University [40]. The first measurements reported
in Ref. [40] are zero field measurements of the quasi-elastic neutron scattering in the (h, h, l)
plane (symmetry equivalent to (h, k, k)). These measurements are shown in Fig. 1.13, along
with one set of data collected in a magnetic field. It can be seen that for the zero field
data, as the temperature decreases from 4 K to 200 mK, rod-like features develop in the
neutron scattering. These rods increase in intensity with decreasing temperature, but, as
the temperature decreases to 30 mK in zero field, the rods begin to break up, bunching up
around the nuclear Bragg peaks, as seen in Fig. 1.13. Rod-like features in neutron scattering
21
Figure 1.10: Figure reproduced from Ref. [30]. Quasi-elastic neutron scattering in the
(h, k, k) plane at T = 1.4 K. The most interesting feature of this data is the [111] rod of
neutron scattering intensity. The red lines are the directions along which Q scans were
performed to determine the temperature dependence of the results.
typically indicate the presence of the two dimensional correlations in planes perpendicular
to the direction of the rods of scattering. In the case of the pyrochlore lattice, the planes
perpendicular to [111] are alternating triangular and kagome planes, as shown in Fig. 1.12.
Broad peaks in the scattering on the other hand indicate three dimensional correlations.
The observed bunching up of the intensity is taken as an indication of a crossover from
two dimensional correlations to three dimensional correlations. The transition is not one
to LRO, as this would cause the formation of magnetic Bragg peaks and not the broad
features seen in the data. Fig. 1.13 also shows data taken at 30 mK in a 2 T field. This
data shows distinct Bragg peaks and no rod-like features whatsoever, indicating that the
field has induced the system in a magnetically well-ordered state.
In addition to quasi-elastic neutron scattering, Ref. [40] reports inelastic neutron scattering along the [hhh] direction in reciprocal space, shown in Fig. 1.14. This data confirms
the existence of magnetic LRO in the low temperature (T = 30 mK), high field region of
the phase diagram of Yb2 Ti2 O7 , through the existence of sharp dispersive spin wave excitations that develop at a field of 0.5 T. The lack of these excitations in the low temperature
phase in zero magnetic field is consistent with a lack of magnetic LRO in this phase.
22
Figure 1.11: Figure reproduced from Ref. [30]. The temperature dependence of the rods of
neutron scattering at selected points in Q space, Q = (1.5, 1.5, 1.5) and Q = (1.9, 1.9, 1.9).
Ref. [40] also presents a phase diagram for Yb2 Ti2 O7 based on neutron scattering and
magnetization measurements consisting of three distinct phases, the high temperature,
small field paramagnetic phase (including the region where rods of scattering appear), the
low temperature, small field 3D correlated phase, and the high field ordered phase. A
statement is made that the high field ordered phase and the low field paramagnetic phase
are distinct phases, and that the high field ordered phase is not simply a field polarized
paramagnet, based on a minimum that appears in the magnetization as a function of the
field strength. Finally, Ref. [40] makes a comment on the scattering intensity of the Bragg
peaks as a function of magnetic field. The intensities of some of the peaks, particularly
(2, 2, 2), are found to decrease as the magnetic field is ramped up to values below that of
the zero field intensity. This seems to suggest some kind of field driven structural phase
transition.
Finally we come to the neutron scattering measurements used in this work. These
measurements were collected on the D23 diffractometer at the ILL in Grenoble, France by
Henrik Rønnow and Louis-Pierre Regnault on a single crystal of Yb2 Ti2 O7 . This data is
a more detailed version of the neutron scattering data reported in Ref. [30], and has been
23
z
z
x
y
(a)
x
y
(b)
Figure 1.12: (a) The cubic unit cell of the pyrochlore lattice showing the alternating kagome
(red) and triangular (blue) planes perpendicular to the [111] cubic body diagonal direction.
(b) A view of the same lattice along the [111] direction [21].
Figure 1.13: Figure reproduced from Ref. [40]. Diffuse quasi-elastic neutron scattering
intensity in the (h, h, l) plane. Bright red dots indicate nuclear Bragg peaks.
24
Figure 1.14: Figure reproduced from Ref. [40]. Inelastic neutron scattering along the [hhh]
direction in reciprocal space, for various strengths of the magnetic field along [11̄0]. Panel
a shows how the (h, h, l) planes is tiled by first Brillouin zones, while Panels b-h show the
inelastic neutron scattering. Panels b and c show the scattering in zero field for T = 4
K and T = 30 mK, while panels d-h show the scattering for T = 30 mK for various field
strengths. It can be seen that for field strengths of greater than 0.5 T, sharp spin waves
appear, clear evidence of magnetic LRO.
reported in Ref. [31], the contents of which makes up part of this study. Two sets of data
are presented in Ref. [31], data at T = 1.4 K, and data at T = 9.1 K, shown in Fig. 1.15.
As can be clearly seen in panels (a) and (b) of Fig. 1.15, a rod of scattering is present in the
data, as was found in Refs. [30, 40] though it is quite weak at T = 9.1 K. Fig. 1.15(c) shows
a cut through the (h, k, k) plane along [h, 2.25, 2.25] for both T = 1.4 K and T = 9.1 K.
This cut shows that the intensity of the scattering feature in the top right corner of panels
(a) and (b) of Fig. 1.15 is not magnetic in nature as it does not evolve with temperature.
The rods of scattering intensity that appear in the neutron scattering collected by
Henrik Rønnow are directly related to the correlations between the magnetic moments of
the Yb3+ ions in Yb2 Ti2 O7 , as discussed in Ref. [40]. These correlations are connected
to the magnetic interactions present in this material. These relationships should make it
possible to determine the form of the magnetic interactions present in Yb2 Ti2 O7 from the
quasi-elastic neutron scattering of Fig. 1.15, specifically the data collected at T = 1.4 K.
A knowledge of the magnetic interactions present in this material is the first step towards
understanding the low temperature behaviour of Yb2 Ti2 O7 , particularly the observed zero
field phase transition and low temperature phase. As we saw in the previous section, previous researchers [16, 19, 20, 33, 34, 37] have attempted to understand the interactions in this
material, starting with simple Heisenberg models [16, 20, 33] and progressing to anisotropic
exchange [19, 34, 37]. Each of these attempts have had shortfalls, either assuming an overly
simplistic form for the exchange exchange interaction as in Refs. [16, 20, 33], or when considering the presence of anisotropic exchange interactions, failing to consider the sublattice
structure or full symmetry of the the material [19, 34, 37]. The work of this study will
25
3200
3
3
3000
2
Intensity
Intensity
4
(h,0,0)
(h,0,0)
4
2
1
1
1.4 K
9.1 K
2800
2600
nonmagnetic
feature
2400
a
0
0
0.5
1
1.5
(0,k,k)
2
2.5
b
0
0
0.5
1
1.5
(0,k,k)
2
2.5
2200
0
c
1
2
Q=(h,2.25,2.25)
3
4
Figure 1.15: Neutron Scattering in the (h, k, k) plane of reciprocal space in zero external
field at (a) T = 1.4 K and (b) T = 9.1 K. Panel (c) shows the neutron scattering along the
line [h, 2.25, 2.25] at T = 1.4 K and T = 9.1 K. The important feature to note is that the
scattering intensity between h = 3 and h = 4, revealing that the feature in the top right
corner of panels (a) and (b) near q = (3.5, 2.25, 2.25) is not magnetic in origin. The white
arrows in (a) and (b) show the range h ∈ [3, 4] along the direction [h, 2.25, 2.25].
be to determine the form of the magnetic interactions in Yb2 Ti2 O7 from the quasi-elastic
neutron scattering data of Fig. 1.15. Once the form of the magnetic interactions has been
determined, we will set out to determine the form of the magnetic correlations driven by
these interactions that lead to the rod-like features in the neutron scattering. We will also
compute other quantities that have been measured in experiment, such as the bulk and
local susceptibilities. Finally we will explore the low temperature behaviour of the system
described by our model Hamiltonian.
26
Chapter 2
The Model Hamiltonian for Yb2Ti2O7
The first step in determining the form of the magnetic interactions in Yb2 Ti2 O7 is to define
a model magnetic Hamiltonian for this material. This Hamiltonian, H, will describe the
local crystal field environment inhabited by the Yb3+ ions in Yb2 Ti2 O7 and the exchange
and long-range magnetic dipolar interactions between the magnetic moments of the Yb3+
ions. We define this Hamiltonian as
H = Hint + HCF ,
(2.1)
where HCF describes the crystal field environment that the Yb3+ ions inhabit, and Hint
describes all of the interactions between the moments of the Yb3+ ions.
2.1
The Crystal Field of Yb2Ti2O7
Yb3+ ions have a 4f 13 , 2 F7/2 electronic configuration. As there are an odd number of
electrons in the valence shell, Yb3+ is a Kramers ion, and thus the crystal field must have
an even number of states. The crystal field of Yb2 Ti2 O7 splits the 2 F7/2 energy level into
four Kramers doublets [20]. As discussed in the previous chapter, the crystal field of Yb2 Ti2 O7 has been probed experimentally by various groups as far back as 1968 [22]. The early
studies of this material [22, 23] failed to use the correct crystal field parameters required
by symmetry in their descriptions of the crystal field. The site symmetry of the pyrochlore
lattice site that the Yb3+ ions inhabit is D3d [20]. For this type of symmetry, the crystal
field must be described by six independent crystal field parameters, so that HCF can be
written as:
(2.2)
HCF = B20 O20 + B40 O40 + B43 O43 + B60 O60 + B63 O63 + B66 O66 ,
27
where Onm are Steven’s operators, which are defined in Appendix B. The oldest work
on the crystal field of Yb2 Ti2 O7 , Ref. [22] considered only three of the six symmetry
required terms (B20 ,B40 , and B60 ), while the next work, Ref. [23], considered only two of the
symmetry allowed terms (B20 and B40 ). Reference [25] considered all independent crystal
field terms, and based on magnetic susceptibility measurements and Mössbauer spectra,
proposed a set set of crystal field levels where the ground state is purely |Jx = ±3/2i,
and the energy gap between the ground state and first excited state is ∆E ∼ 138 K. This
crystal field parameterization is inconsistent with the findings of more recent experiments
in Refs. [19, 20]. These two works propose similar crystal field schemes that disagree with
the findings of Ref. [25].
We will use two different crystal field (CEF) parameterizations for Yb2 Ti2 O7 in our
attempts to extract the form of the magnetic interactions in this material, that of Hodges
et al. [20], discussed in Appendix A.1, and that of Cao et al. [19], discussed in Appendix
A.2. Both of these CEF parameterizations yield a set of CEF states where the ground state
doublet is well separated from the excited states and has a local easy plane anisotropy.
2.2
Magnetic Interactions
To describe the magnetic interactions between the magnetic moments of the Yb3+ ions we
propose an interaction Hamiltonian
Hint = Hex + Hdip ,
(2.3)
where Hex describes all of the symmetry allowed nearest-neighbour bilinear exchange interactions for the pyrochlore lattice, and Hdip describes the long-range magnetic dipolar
interaction. We choose to restrict the exchange interactions in this model Hamiltonian to
bilinear exchange based on the fact that the energy gap between the ground state doublet
and the lowest excited state doublet of the CEF, ∆ ∼ 650 ± 40 K, [20, 19] is much larger
than the temperature at which the quasi-elastic neutron scattering data was collected, 1.4
K. Also the strength of the magnetic dipolar and exchange interactions in this material,
an indication of which is given by θCW = 0.70 ± 0.05 [18, 20], is much less than that of the
CEF splitting. Because both the interaction energy and temperature are much less than
the CEF splitting, we can safely project the microscopic interaction Hamiltonian, Hmic ,
which may contains higher order multipolar terms because Yb3+ is a J = 7/2 ion, onto a
low energy Hilbert space spanned only by the direct product of the states in the ground
state doublet of the CEF [31]. This will allow us to consider only an effective Hamiltonian
Heff that describes bilinear exchange interaction between effective spin-1/2 doublets, where
28
the bilinear interactions between these spin-1/2 operators represent the “down-projection”
of higher order multipolar magnetic interactions between the full J angular momentum
operators.
2.2.1
Symmetry Allowed Bilinear Exchange Interactions
The symmetry allowed nearest-neighbour exchange interactions for the pyrochlore lattice are derived and discussed in Appendix C. There are four such interactions {Xn } =
X1 , X2 , X3 , X4 . X1 and X2 contain the components of the conventional Ja · Jb = Jax Jbx +
Jay Jby + Jaz Jbz Heisenberg exchange interaction, split into two terms. X1 contains two of
the three terms that make up the Heisenberg exchange interaction, and X2 contains the
third term. The specific bilinear exchange terms contained in X1 and X2 vary depending
on the sublattices as shown in Eqns. (C.7) and (C.8). X4 (Eqn. (C.10)) contains all of the
anti-symmetric symmetry allowed bilinear exchange terms, and X3 (Eqn. (C.9)) contains
all of the remaining bilinear exchange terms not contained in X1 , X2 , and X4 . Together
these four terms span all of the possible bilinear exchange terms for a pyrochlore material. The energy associated with each of the symmetry allowed exchange interactions
{Jn } = {J1 , J2 , J3 , J4 } will be determined by fitting to the neutron scattering data provided by Prof. Rønnow and collaborators. The method used to accomplish this will be
discussed later in Chapter 4. Hex is defined as
Hex =
X
(Jai )T J
a,b
(i, j) Jbj
(2.4)
hi,a;j,bi
where Jai and Jbj are the classical angular momentum three-vectors at the i and j FCC
lattice sites, and the a and b tetrahedral sublattice sites defined in Chapter 1.2. J a,b (i, j) =
NN
NN
is a 3 × 3 matrix that describes the interactions between Jai and Jbj , where δi,a;j,b
J a,b δi,a;j,b
imposes that the pyrochlore lattice sites indexed by the FCC lattice labels i, j and sublattice
labels a, b are nearest-neighbours. The angular momentum vectors Jai and Jbj are written
in cartesian coordinates. We have chosen to use the full J angular momentum operators
so that the RPA method of calculating the neutron scattering, described in Ref. [42] and
discussed in Chapter 3.3 can be utilized. This means that the values of {Jn } that are
determined from fitting to experiment will be effective “unprojected” couplings, whose
projection on to the ground state doublet of the CEF correctly describes the low energy
physics of Yb2 Ti2 O7 . They are likely not microscopic couplings that correctly describe the
physics of this material at energy scales on the order of the CEF splitting [31]. This makes
29
our model effectively an “un-projected” form of spin-1/2 Hamiltonian [31]. The relationship
between our model and effective spin-1/2 models is discussed further in Appendix H.
a,b
To aid in understanding, the matrices J are most easily thought of as component
matrices of a larger 12 × 12 matrix, J , where a and b index the positions of the 3 × 3
submatrices within J . That is,

1,2
1,3
1,4 
0 J
J
J
 2,1
2,3
2,4 

 J
0
J
J


(2.5)
J =  3,1
3,2
3,4  .

 J
J
0
J


4,1
4,2
4,3
J
J
J
0
The nearest-neighbour condition on the interactions adds the constraints that the submatrices where a = b must be zero. This is illustrated in Fig. 2.1, where it can be seen that for
the pyrochlore lattice, there are no same-sublattice nearest-neighbours. Symmetry imposes
T
b,a
a,b
the condition that J = J
.
4
1
2
3
2
3
4
Figure 2.1: An illustration of the nearest-neighbours of sublattice site a = 1 of the pyrochlore lattice. Here we have chosen sublattice a = 1 as the origin to illustrate that none
of the nearest-neighbours of this sublattice site are also a = 1 sites. This applies equally
for all of the other sublattice sites a = 2, 3, 4.
For our model of symmetry allowed nearest-neighbour bilinear exchange, J consists
of a linear combination of the matrix forms of the symmetry allowed nearest-neighbour
30
bilinear exchange interactions as defined in Eqn. (C.15)
J = −J1 X1 − J2 X2 − J3 X3 − J4 X4 .
(2.6)
The chosen sign convention for the Hamiltonian means that if J1 and J2 are positive, the
interaction associated with these two terms will be ferromagnetic in nature.
2.2.2
Alternative Representations of the Symmetry Allowed Bilinear Exchange Interactions
In order to perhaps ascribe a simpler and seemingly more physical meaning to the couplings
{Jn }, it is useful to define a more physically motivated representation of the bilinear
exchange interactions. This is done by taking a linear combination of the symmetry allowed
exchange terms Xn , as explained in Appendix C.2. This allows us to write the component
of the Hamiltonian containing the bilinear exchange terms, Hex , as
Hex = HIsing + Hiso + Hpd + HDM .
X
HIsing = −JIsing
(Jai · ẑa ) Jbj · ẑb
(2.7)
(2.8)
<i,a;j,b>
is the nearest-neighbour local Ising exchange, which couples the local ẑ components of Jai
on neighbouring sublattice sites.
X
Hiso = −Jiso
Jai · Jbj
(2.9)
<i,a;j,b>
is nearest-neighbour Heisenberg exchange, and
X
b
ab
(Jai · Jbj − 3(Jai · R̂ab
Hpd = −Jpd
ij )(Jj · R̂ij ))
(2.10)
<i,a;j,b>
is the nearest-neighbour pseudo-dipolar exchange, an exchange interaction of the same
form, but not related to Hdip .
X
a
b
(2.11)
HDM = −JDM
Ωa,b
DM · Ji × Jj
<i,a;j,b>
is the Dzyaloshinskii-Moriya (DM) exchange on the pyrochlore lattice (See Chapter C.2 for
details). We denote the set of four exchange energies associated with this representation of
the symmetry allowed bilinear exchange interactions {Je } = {JIsing , Jiso , Jpd , JDM }. This
representation of the symmetry allowed bilinear exchange interactions will be used in the
interpretation of the couplings {Jn } determined from fits to the experimental quasi-elastic
neutron scattering in Chapter 4.
31
2.2.3
The Long-Range Magnetic Dipolar Interaction
The final term in Hint is Hdip , which describes the long-range magnetic dipolar interactions.
Hdip is defined as
Hdip = D
3
rnn
b
ab
ij )(Jj · R̂ij )),
ab 3
|R
|
ij
i>j;a,b
X
(2.12)
√
(gJ µB )2
ab
is
defined
in
Eqn.
(1.2),
and
r
=
≈
0.01848
K,
R
2rc /4, where
where D = µ04πr
nn
3
ij
nn
rc is the cubic unit cell dimension, defined in Chapter 1.2. µ0 is the permeability of free
space, µB is the Bohr magneton, and gJ = 8/7 is the Landé factor for Yb3+ . This term
describes the interaction between the dipole moments of the Yb3+ ions via magnetic fields
rather than exchange. Computing the sum over all of the magnetic moments Jai is done
using the Ewald sum, which is discussed in detail in Appendix E.
2.3
Summary
In this chapter we have defined the magnetic Hamiltonian that we will use to describe Yb2 Ti2 O7 in this study. This model consists of the local crystal field environment inhabited by
the Yb3+ ions as well as interactions between the magnetic moments of the Yb3+ ions. To
describe the interactions between the magnetic moments, we have proposed a combination
of all of the symmetry allowed nearest-neighbour bilinear exchange interactions on the
pyrochlore lattice combined with long-range magnetic dipolar interactions. This model
does not specify the strengths of the various bilinear exchange terms, so a method is
required to determine the strengths of the various interactions. Such a method will be
discussed in Chapter 4.
32
Chapter 3
Magnetic Neutron Scattering
In this chapter we will explore the microscopic origins of magnetic neutron scattering in
order to understand how neutrons probe the magnetic properties of a material. We will
also derive the relationship between the quasi-elastic neutron scattering cross-section and
the wave-vector dependent magnetic susceptibility, χ (q), and describe the random phase
approximation method of calculating χ (q). These will be important tools for determining
the strengths of the symmetry allowed bilinear exchange terms in our model of Yb2 Ti2 O7
discussed in Chapter 2.
3.1
The Microscopic Origin of Neutron Scattering
In this section we derive the equation for the neutron scattering cross section starting from
the microscopic neutron-scatterer interaction Hamiltonian. The scatterer is the material
upon which the neutrons are incident, and which is responsible for the scattering of the
neutrons through various interactions, hence its name. Our derivation of the microscopic
origin of the neutron scattering cross section will follow that of Jensen and Mackintosh,
Ref. [41]. This formulation for the scattering cross section is based on the dipole approximation. The resulting form of the neutron scattering will be used in combination with
the random phase approximation discussed in the next section to compute neutron scattering from our model magnetic interactions Hamiltonian for Yb2 Ti2 O7 for comparison to
experiment.
In a neutron scattering experiment a beam of neutrons interacts with the material
sample (the scatterer) via two mechanisms, the nuclear force, and interaction between
33
the neutron’s magnetic moment with the magnetic field of the electrons in the sample. In
materials with unpaired electrons such as Yb2 Ti2 O7 , these two terms yield contributions of
roughly equal magnitude to the neutron scattering cross section [41]. We are not interested
in the crystalline structure of Yb2 Ti2 O7 , so we will neglect the nuclear force as it leads only
to structural Bragg peaks and does not tell us anything about the magnetic interactions in
the sample. The derivation in Ref. [41] that we follow in this section considers the magnetic
field from the electrons in the scatterer using a multipole expansion. The first term of this
expansion, the dipolar term, provides the dominant contribution to the scattering at small
scattering wave-vectors.
The neutron scattering process begins with a collimated monochromatic beam of neutrons with equal energy. We define the plane wave initial (“pre-scattering”) state of the
neutrons as [41]
|ksn i = V −1/2 exp (ik · rn ) |sn i,
(3.1)
where k is the wave-vector of the neutron, |sn i is the spin state of the spin-1/2 neutron,
2
where
and V is the volume of the system. The energy of the neutron beam is given by ~k
2M
M is the mass of the neutron. Using Fermi’s golden rule, we can determine the probability
that the neutron undergoes a transition from its initial state |ksn i to the state |k0 s0n i, which
is given by (Eqn. 4.1.1 in Ref. [41])
2π X
W ksn , k0 s0n =
Pi |hksn ; i|Vint |k0 s0n ; f i|2 δ (Ek − Ek0 + Ei − Ef ) ,
~
(3.2)
i,f
where Pi is the probability that the scatterer is initially in a state |ii, Vint is the interaction
potential that describes the interaction between the neutron and the scatterer, and |f i is
the final state of the scatterer. Ek and Ek0 are the initial and final energies of the neutron,
such that Ek − Ek0 = ~ω, and Ei and Ef are the energies of the initial and final states of
the Hamiltonian of the scatterer, that is H|ii = Ei |ii and H|f i = Ei |f i. The sum is over
all final and initial states. Information about the scatterer is extracted by measuring the
energy transfer ~ω and momentum transfer ~Q of the neutron, where Q = k − k0 is the
scattering wave-vector [41].
The basic formulation of the neutron scattering cross section in terms of experimentally
controllable or measurable parameters is defined in Eqn. (1.6). In terms of the microscopic
interaction of the neutron with the scatterer [41]
δN =
V M |k0 |
V
0 2
0
(k
)
d|k
|dΩ
=
dEdΩ,
(2π)3
(2π)3 ~2
(3.3)
is the number of neutrons within the energy range dE, directly related to the wave-vector
of the neutrons, and solid angle range (corresponding to a range of k0 /|k0 |) dΩ. The flux,
34
j (k, sn ) of inbound neutrons with wave-vector k and spin sn is given by ~k/ (V M ). Using
these definitions, Eqn. (1.6) yields (Eqn. 4.1.4a in Ref. [41])
d2 σ
|k0 |
=
dΩdE
|k|
M
2π~2
2 X
Pi |hsn ; i|Vint (Q) |s0n ; f i|2 δ (Ek − Ek0 + Ei − Ef ) ,
(3.4)
i,f
where Vint (Q) is the Fourier transform of the neutron-scatterer interaction potential (Eqn.
4.1.4b in Ref. [41])
Z
Vint (Q) =
Vint exp (iQ · rn ) drn ,
(3.5)
where rn is the position of the neutron.
Now we can move on to defining the neutron-scatterer interaction potential Vint . The
magnetic interaction between the magnetic moment of a neutron and the magnetic field due
to a single electron with momentum p is is described by the potential, in CGS-Gaussian
units, (Eqn. 4.1.5 in Ref. [41])
2
e
1 e 2
1 p + (An + Ae ) −
p + Ae + 2µB s · Bn
2m c
2m
c
1
= 2µB
An · p0 + s · (∇ × An ) ,
~
Vint (re , rn ) =
(3.6)
where re and rn are the position of the electron and neutron respectively, p is the momentum of the electron, and An is the magnetic vector potential due to
the magnetic moment
1
of the neutron µn , such that Bn = ∇ × An = ∇ × −µn × ∇ r . Ae is the additional
contribution to the total magnetic vector potential from the surrounding electrons in the
scatterer or an external field. s is the spin of the electron, µB = 9.274 × 10−24 J/T is the
Bohr magneton, e is the absolute value of the charge on an electron, and c is the speed
of light. p0 = p + ec Ae is the adjusted electron momentum. The Fourier transform of An ,
with respect to the neutron coordinate is given by [41]
Z
4π
An (re − rn ) exp (iQ · rn ) drn = − exp (−iQ · re )
µn × Q̂,
(3.7)
i|q|
where µn is the magnetic moment of the neutron, and Q̂ is a unit vector in the direction
of Q. The Fourier transform of ∇ × An is given by [41]
Z
∇ × An (x) exp (iQ · x) dx = 4π Q̂ × µn × Q̂.
(3.8)
35
Combining the results of Eqn. (3.7) and Eqn. (3.8) with Eqn. (3.6), we obtain (Eqn. 4.1.6
in Ref. [41])
Z
Vint (Q) = Vint (re − rn ) exp (iQ · rn ) drn
i
0
Q̂ × p + Q̂ × s × Q̂ exp (−iQ · re )
(3.9)
= 8πµB µn ·
~|Q|
In the case of rare earth ions, such as Yb2 Ti2 O7 , we can impose a further restriction on
the interaction between the neutron and electron: that the electrons are localized around
the Yb3+ lattice sites, thus re = Rai +r. This allows us to rewrite Eqn. (3.9) as (Eqn. 4.1.7a
in Ref. [41])
Vint (Q) = 8πµB µn · (Zp + Zs ) exp (−iQ · Rai ) ,
(3.10)
where (Eqn. 4.1.7b in Ref. [41])
i
Q̂ × p0 exp (−iQ · r)
~|Q|
(3.11)
Zs = Q̂ × s × Q̂ exp (−iQ · r) .
(3.12)
Zp =
and
In order to compute the matrix elements hi|Zp,s |f i, it is explained in Ref. [41], that
we must expand the the factor exp (−iQ · r) in Zp and Zs in terms of Bessel functions
jn (ρ), where ρ = |Q||r|, and cos(θ) = Q · r/ρ. This expansion has the form (Eqn. 4.1.8 in
Ref. [41])
∞
X
exp(−iQ · r) =
(2n + 1)(−i)n jn (ρ)Pn (cos(θ)),
(3.13)
n=0
where Pn is nth Legendre polynomial. If we perform this expansion for Zp , we obtain the
result [41]:
i
1
0
0
j0 (ρ)p + {j0 (ρ) + j2 (ρ)} Q̂ · r p + . . . .
(3.14)
Zp = Q ×
~|Q|
~
This result can then be rearranged and expressed as (Eqn. 4.1.9a in Ref. [41])
Zp =
1
{j0 (ρ) + j2 (ρ)} Q̂ × l0 × Q̂ + Z0p ,
2
36
(3.15)
Z0p
= Q̂ ×
i
1
j0 (ρ)p0 + {j0 (ρ) + j2 (ρ)}{(Q̂ · r)p0 − (Q̂ · p0 )r} + . . .
~|Q|
2~
(3.16)
contains all of the remaining terms in the expansion of Zp . ~l0 = ~l + ec r × Ae , where
l = r×p
is the angular momentum. Ref. [41] shows that because p0 = im
[H, r], all of the
~
~
terms in Z0p can be expressed in terms of commutators with H so that (Eqn. 4.1.10 in
Ref. [41])
Z0p =
m
i|Q|
Q̂
×
−j
(ρ)[H,
r]
+
{j
(ρ)
+
j
(ρ)}[H,
Q̂
·
r
r]
+
.
.
.
.
0
0
2
~2 |Q|
2
(3.17)
For any operator Â, hi|[H, Â]|f i = (Ei −Ef )hi|Â|f i, which means that in the limit Q → 0,
this term will not contribute to hi|Zp |f i, because Ef → Ei as Q → 0.
The expansion of Zp has shown that the leading term in Zp is of the form Q̂ × Al0 × Q̂,
where A = 21 {j0 (ρ) + j2 (ρ)}, which is the same general form as that of Zs in Eqn. (3.12).
This allows us to define the new vector operator K (Q) such that (Eqn. 4.1.11 in Ref. [41])
Q̂ × K × Q̂ = Zp + Zs .
In the limit Q → 0 (Eqn. 4.1.12a [41])
e
2µB K (0) = µB l + r × Ae + 2s = −µe ,
~c
(3.18)
(3.19)
where µe is the magnetic dipole moment of the electron. This yields the result (Eqn. 4.1.12b
[41])
Vint (0) = −4πµB · Q̂ × µe × Q̂ .
(3.20)
In the case where Q 6= 0, an analytic solution for Vint (Q) is almost impossible to obtain
because Z0p cannot be neglected and jn (ρ) do not commute with H [41]. However, if the
scattering processes are restricted to those where the orbital quantum number l of the
electron is conserved, then the first term in Eqn. (3.17) vanishes identically because j0 (ρ)
and H are diagonal in l. In the second term of Eqn. (3.17), it is possible to replace H with
only the kinetic term of the Hamiltonian to first order if l is conserved. It is then shown
in Ref. [41], that if the principal quantum number n and the angular momentum quantum
number l are conserved between |ii and |f i states, the electron wave-function is radially
symmetric and the second term in Eqn. (3.17) also vanishes identically [41]. If n is not
conserved, or if H is not diagonal in l, then this term will lead to an imaginary contribution
37
to K (Q), and a contribution to the neutron scattering cross section proportional to Q2 ,
however this contribution is normally very small [41].
The assumption that n and l are conserved between the |ii and |f i, so that the two
highest order terms in Eqn. (3.17) are negligible means that Kp , the portion of K(Q)
arising from Zp is given by (Eqn. 4.1.13a in Ref. [41])
Kp (Q) =
1
{hj0 (|Q|)i + hj2 (|Q|)i} l.
2
hjn (|Q|)i is defined as (Eqn. 4.1.13b in Ref. [41])
Z ∞
r2 R2 (r)jn (|Q|r)dr,
hjn (|Q|)i =
(3.21)
(3.22)
0
where R(r) is the normalized radial electron wave-function [41].
The spin contribution to K(Q), Ks (Q) is given by Ks (Q) ≈ hj0 (|Q|)is, neglecting
terms of order greater than n = 2 [41]. Adding the two contributions to K(Q) yields
(Eqn. 4.1.14 in Ref. [41])
1
1
K(Q) = hj0 (|Q|)i(l + 2s) + hj2 (|Q|)il.
2
2
(3.23)
This P
result can bePgeneralized to an ion with more than one electron using the relations
L = l and S =
s to yield (Eqn. 4.1.15a in Ref. [41])
1
1
1
K (Q) = hj0 (|Q|)i(L + 2S) + hj2 (|Q|)iL = gJ F (|Q|)J.
(3.24)
2
2
2
gJ is the Landé factor, F (|Q|) = hj0 (|Q|)i+ g2J − 1 hj2 (|Q|)i is the magnetic form factor,
and J is the angular momentum operator of the ion the neutron is interacting with. The
magnetic form factor for Yb3+ is given in Appendix F.
We now return to Eqn. (3.10), where, substituting in Eqn. (3.18), the form of K (Q)
from Eqn. (3.24), and taking the sum over all of the Yb3+ lattice sites, we obtain [41]
Vint (Q) = 8πµB
X 1
i,a
2
gJ F (|Q|) µn · Q̂ ×
Jai
× Q̂ exp (−iQ · Rai ) ,
(3.25)
where Jai is the angular momentum operator for the ion at lattice site i and sublattice site
a. The sum over the Yb3+ lattice sites takes us from the single ion scattering problem to
the crystal scattering problem.
38
Now that we have determined the form of the interaction potential between the neutron
and the scatterer, we can return our attention to the neutron scattering cross section. We
now wish to compute the matrix element squared term in Eqn. (3.4). This term can be
written as [41]
|hksn ; i|Vint (Q) |k0 s0n ; f i|2 = hsn ; i|Vint (Q) |s0n ; f ihs0n ; f |Vint (−Q) |sn ; ii.
(3.26)
In this study, we are only interested in unpolarized neutron scattering, so we assume an
equal distribution of neutrons with spin up and spin down. Therefore, [41]
X
Ps hsn |µn · Q̂ × Jai × Q̂ |s0n ihs0n |µn · Q̂ × Jbj × Q̂ |sn i
sn ,s0n
=
2 1
gn µN
Q̂ × Jai × Q̂ · Q̂ × Jbj × Q̂
2
2 X 1
=
gn µN
δu,v − Q̂u Q̂v Jia,u Jjb,v ,
2
u,v
(3.27)
(3.28)
where gn = 3.827 [41] is the nuclear g factor, and µN is the nuclear magneton, such
that µn = −gn µN sn . u and v are cartesian components. This result gives us one of the
key aspects of neutron scattering measurements: that they measure only the correlations
between the components of the magnetic moments perpendicular
to the scattering wavevector Q. This condition is imposed by the term δu,v − Q̂u Q̂v . This arises from the
form of the Fourier transform of the interaction between the moment of the neutron and
magnetic field due to the electrons, specifically the term involving the curl of the vector
potential of the field due to the magnetic dipole moment of the neutron in Eq. (3.6).
Combining Eqn. (3.28) with Eqn. (3.4), we obtain the final form of the neutron scattering cross section (Eqn. 4.1.16 in Ref. [41])
2
X1
d2 σ
|k0 | ~γe2 X 1
=
δu,v − Q̂u Q̂v
gJ F (|Q|)
gJ F (|Q|)
dΩdE
|k| mc2
2
i,a 2
j,b
u,v
i,a;j,b
X
×
Pi hi|Jia,u exp (−iQ·Rai ) |f ihf |Jjb,v exp iQ·Rbj |iiδ (Ek −Ek0 +Ei −Ef ) , (3.29)
i,f
where γ = e2 /mc2 is the gyromagnetic ratio, and m is the mass of the electron.
Next we examine the integral form of the δ function [41]
δ (Ek − Ek0 + Ei − Ef ) =
1
2π~
Z
∞
exp (i (Ek − Ek0 + Ei − Ef ) t/~) dt.
−∞
39
(3.30)
We can use this formulation of the δ function to write [41]
X
Pi hi|Jia,u |f ihf |Jjb,v |iiδ (Ek − Ek0 + Ei − Ef )
i,f
X 1 Z ∞
dt exp (iωt) Pi hi| exp (iHt) Jia,u exp (−iHt) |f ihf |Jjb,v |ii
=
2π~ −∞
i,f
Z ∞
X
1
=
dt exp (iωt)
Pi hi|Jia,u (t) Jjb,v (0) |ii
2π~ −∞
i
Z ∞
1
dt exp (iωt) hJia,u (t) Jjb,v (0)i
=
2π~ −∞
(3.31)
where we have used the fact that Ek − Ek0 = ~ω. H is the Hamiltonian of the scatterer,
and Jia,u (t) is the angular momentum operator for the ion at lattice site i sublattice site
a in the Heisenberg picture. Then at thermal equilibrium, combining Eqn. (3.31) with
Eqn. (3.29), we can write (Eqn. 4.2.1 in Ref. [41])
d2 σ
|k0 |
=
dΩdE
|k|
~γe2
mc2
2 X
1
gJ F (|Q|)
gJ F (|Q|)
(δu,v − q̂u q̂v )
2
2
i,a
j,b
u,v
i,a;j,b
Z ∞
1
×
dt exp (iωt) exp iQ · Rai − Rbj hJia,u (t) Jjb,v (0)i, (3.32)
2π~ −∞
X1
where we have neglected thermal fluctuations of the positions of the ions, Rai , because
we fix the ions to be at the sites Rai and do not allow for any fluctuations in these positions. If we were to include such fluctuations by letting Rai → R̃ai = Rai + uai , where
uai issome small
ion at site Rai then we can write [41]
fluctuation in the position of the
hexp R̃ai − R̃bj i = exp (−2W (Q)) exp Rai − Rbj , where W (Q) is the Debye-Waller factor. This new exponential term would then appear in Eqn. (3.32). For a further discussion
of the Debye-Waller factor, see Ref. [41].
If the magnetic form factor of all of the ions are the same, which is the case for Yb2 Ti2 O7 , we can take the form factor outside the sum (Eqn. 4.2.2a in Ref. [41])
d2 σ
|k0 |
=N
dΩdE
|k|
~γe2
mc2
2 X
u,v
2 X
1
uv
Sab
(Q, ω) ,
(δu,v − q̂u q̂v ) gJ F (|Q|)
2
(3.33)
a,b
uv
where Sab
(Q, ω) is the Van Hove scattering function (Eqn. 4.2.2b in Ref. [41])
uv
Sab
1
(Q, ω) =
2π~
Z
∞
exp (iωt)
−∞
1 X
exp iQ · Rai − Rbj hJia,u (t) Jjb,v (0)idt,
N
i,j
40
(3.34)
which is simply 1/ (2π~) times the Fourier transform of the correlation function
hJia,u (t) Jjb,v (0)i [41].
3.2
The Relationship Between Quasi-elastic Neutron
Scattering and the Q-dependent Magnetic Susceptibility
We saw in the previous section that in order to compute the neutron scattering cross section
uv
a method for computing the van Hove scattering function, Sab
(Q, ω), is required. In this
uv
section we will show the relationship between Sab
(Q, ω) and the wave-vector dependent
magnetic susceptibility in the static approximation.
We start with the generalized susceptibility, χBA (ω), a measure of the change of an
ensemble averaged physical observable B̂ to a perturbation by some operator Â. The
generalized susceptibility has two components
χAB (ω) = χ0AB (ω) + iχ00AB (ω) ,
(3.35)
where χ0AB (ω) is referred to as the reactive part of χAB (ω) and χ00AB (ω) is referred to as
the absorptive part of χAB (ω). If the matrix elements hi|B̂|f i and hf |Â|ii are real, then
χ0AB (ω) and χ00AB (ω) are the real and imaginary parts of χAB (ω) respectively. χ0AB (ω) is
given by (Eqn. 3.3.6a in Ref. [41])
Ei 6=Ef
χ0AB
(ω) =
X hi|B̂|f ihf |Â|ii
(ni − nf ) + χ0AB (el) δω=0 ,
Ei − Ef − ~ω
if
(3.36)

χ0AB (el) = β 

Ei =Ef
X
hi|B̂|f ihf |Â|iini − hBihAi .
(3.37)
if
β = kB1T , and nα = exp (−βEα ) /Z is the thermal occupation number of state |αi, and Z
is the partition function. χ00AB (ω) is given by (Eqn. 3.3.5 in Ref. [41])
X
hi|B̂|f ihf |Â|ii (ni − nf ) δ (~ω − (Ei − Ef )) .
(3.38)
χ00AB (ω) = π
if
41
To obtain the wave-vector dependent magnetic susceptibility, we take the generalized susceptibility and choose
X a,u
Ji exp (−iQ · Rai )
Â = N −1/2
i
and
B̂ = N −1/2
X
Jjb,v exp iQ · Rbj ,
j
to create the quantity χu,v
a,b (Q, ω).
In the previous section, we showed that the neutron scattering cross section is prodσ
uv
portional to the van Hove scattering function, dΩ
∝ Sab
(Q, ω). This is simply a Fourier
transformed two operator correlation function, where the operators are Jia,u and Jjb,v , which
correspond to our choice of Â and B̂ to define the wave-vector dependent magnetic susceptibility. According to Ref. [43], in the case of paramagnetic scattering near a phase
transition we can write (Eqn. 13.17 in Ref. [43])
1
u,v
2 χa,b (Q, ω)
(gJ µB )
uv
× ω (1 − (exp(−β~ω)))−1 Fab
(Q, ω),
uv
uv
Sab
(Q, ω) − Sab,Bragg
(Q, ω) ∝
(3.39)
uv
where Sab,Bragg
(Q, ω) is the Bragg scattering due to any finite value of hJia,u i, which is
uv
very small in the paramagnetic phase and contributes only at ω = 0 [43]. Fab
(Q, ω) is the
spectral weight function, which we will discuss shortly. This equation has been modified
from the simple case of Ref. [43] to include sublattice structure. Equation (3.39) arrises
from the fact that (Eqn. 13.9 in Ref. [43])
uv
u
v
Sab
(Q, ω) = ω (1 − exp (−~ωβ))−1 Ruv
ab,Q (ω) + δ(~ω)hJa ihJb i,
(3.40)
uv
where Ruv
ab,Q (ω) is the Fourier transform of the relaxation function Rab,Q (t), which describes
the relaxation of the magnetization due to a discontinuous magnetic field [43]. From this,
it can be shown that (Eqn. 13.11 in Ref. [43])
Z ∞
1 − exp(−~ωβ)
uv
uv
Sab
(Q, ω) − Sab,Bragg
= Ruv
(3.41)
dω
ab,Q (t = 0),
ω
−∞
u,v
N
and from Eqn. 13.12 of Ref. [43], Ruv
ab,Q (t = 0) = (gj µb )2 χa,b (Q, ω). Using this relation
u,v
between Ruv
ab,Q and χa,b (Q, ω) then allows us to invert Eqn. (3.41) to obtain Eqn. (3.39).
The spectral weight function F (Q, ω) is included in Eqn. (3.39) to account for the energy
42
dependence of the inelastic neutron scattering intensity, and is defined as (Eqn. 13.15 in
Ref. [43])
!
Z ∞
uv
(t)
R
1
ab,Q
uv
dt exp (−iωt)
(3.42)
Fab
(Q, ω) =
2π −∞
Ruv
ab,Q (t = 0)
So far we have considered the fully general equation for the energy dependent neutron
scattering, Eqn. (3.33), which contains both the elastic and inelastic neutron scattering.
The inelastic scattering is the scattering where energy of the scattered neutrons is not
conserved, while elastic scattering is scattering where the energy of the neutrons is exactly
conserved. In order to obtain the quasi-elastic portion of the neutron scattering cross
section we use the static approximation.
The static approximation states that if the difference in energy of the incoming and
scattered neutrons is sufficiently small the conservation of energy in Eqn. (3.4) can be
neglected. This disregard for conservation of energy is allowed due to lack of fine energy
resolution in the detector and the fact that the energy spectrum of the states of the scatterer
involved in the response to the neutron is bounded, with a bandwidth much less than the
energy of the incoming neutrons. More explicitly, the static approximation assumes that
Ek Ei and Ek0 Ef , so that δ (~ω − (Ei − Ef )) ≈ δ(~ω) [44]. This means that
Ek ≈ Ek0 and |k0 |/|k| ∼ 1 [44]. This approximation allows us to then simply integrate
over dE in Eqn. (3.29) removing the energy dependence of dσ 2 /dEdΩ and removing δ(~ω)
leaving the rest of Eqn. (3.29) [44]. This means that all information about the time
dependence of the correlations in Eqn. (3.32) is lost, which is why it is called the static
approximation. The static approximation also involves the approximation that for small
ω, (1 + exp(−β~ω))−1 ≈ (β~ω)−1 , giving us
uv
uv
Sab
(Q, ω = 0) − Sab,Bragg
(Q, ω = 0) ∝ kB T
1
u,v
2 χa,b (Q, ω = 0) ,
(gJ µB )
(3.43)
where the proportionality now includes the ω = 0 value of the spectral weight function
F (Q, ω). Finally, we can place this result into our equation for the neutron scattering cross
section to obtain
X
X u,v
dσ
∝ kB T |F (|Q|)|2
χa,b (Q) ,
dΩ
u,v
a,b
(3.44)
the quasi-elastic neutron scattering cross section in terms of the wave-vector dependent
magnetic susceptibility in the static approximation. This result is in agreement with
43
Ref. [45]. In order to maintain consistency with Ref. [42], we choose to express χu,v
a,b (Q) in
u,v
terms of χa,b (q). This is done by expanding the operators
X a,u
Ji exp (−iQ · Rai )
Â = N −1/2
i,a
and
B̂ = N −1/2
X
Jjb,v exp iQ · Rbj
j,b
as
Â = N −1/2
X
Jia,u exp (−iQ · Rai )
i,a
=
X
exp (−iG · ra ) Jia,u exp (−iq · Rai )
(3.45)
i,a
B̂ = N −1/2
X
Jjb,v exp iQ · Rbj
j,b
=
X
exp (iG · rb ) Jia,u exp iq · Rbj .
(3.46)
i,a
This allows us to write:
a
b
χu,v
· G χu,v
a,b (Q) = exp −i r − r
a,b (q) ,
(3.47)
where χu,v
a,b (q) is now the generalized susceptibility for the operators
X a,u
Â = N −1/2
Ji exp (−iq · Rai )
i
and
B̂ = N −1/2
X
Jjb,v exp iq · Rbj .
j
This gives us the result
X
X
dσ
∝ kB T |F (|Q|)|2
exp −i ra − rb · G χu,v
a,b (q) .
dΩ
u,v
a,b
(3.48)
dσ
Ref. [42] reports a different temperature dependence for dΩ
than is given in Eqn. (3.48),
that is (Eqn. 2 in Ref. [42])
X
X
dσ
1
∝
|F (|Q|)|2
exp −i ra − rb · G χu,v
(3.49)
a,b (q) .
dΩ
kB T
u,v
a,b
44
This temperature dependence is only an over all scaling factor on the neutron scattering
that will be cancelled out later in the process of determining the form of the magnetic
interactions in Yb2 Ti2 O7 , so we do not give it further consideration, but it should be noted
that Eqn. (3.49) is the form used all calculations in this work.
3.3
The Random Phase Approximation Formula for
Quasi-elastic Neutron Scattering
Now that we can compute the quasi-elastic neutron scattering cross section directly from
the magnetic susceptibility, we require only a way to compute this quantity in terms of the
CEF states of Yb2 Ti2 O7 and our magnetic interaction Hamiltonian, Hint . To accomplish
this we choose the random phase approximation (RPA). The random phase approximation
arises from linear response theory, which can be used to show that many experimentally
observable quantities, including the neutron scattering cross section, can be expressed
in terms of two-particle correlation functions [41]. The RPA method of computing the
magnetic susceptibility is discussed in Ref. [41]. The starting point for the RPA method
of calculating the magnetic susceptibility is Eqn. (2.1)
H = Hint + HCF
X
a,b
=
(Jai )T K (i, j) Jbj + HCF ,
(3.50)
(3.51)
hi,a;j,bi
where K(i, j) is the magnetic interaction matrix, the elements of which are given by
a,b
a,b
a,b
Ku,v
(i, j) = Ju,v
(i, j) + Du,v
(i, j) .
(3.52)
a,b
(q) are the elements of the interaction matrix J given by Eqn. (2.5) of Chapter 2,
Ju,v
a,b
and Du,v
(i, j) are the elements of the real space dipolar interaction matrix [46, 47]. The
next step is to introduce the thermal expectation values of the the angular momentum
operators hJai i, and rewrite the Hamiltonian as (Eqn. 3.5.2 in Ref. [41])
X
a,b
1 X a
H=
HMF (i, a) −
(Ji − hJai i)T K (i, j) Jbj − hJbj i ,
(3.53)
2 i,a6=j,b
i,a
where the sum over i, a 6= j, b indicates a sum over all lattice, sublattice site pairs that are
not equivalent. HMF (i, a) is defined as (Eqn. 3.5.3 in Ref. [41])
T
X
a,b
1 a
a
HMF (i, a) = HCF (i, a) −
Ji − hJi i
(3.54)
K (i, j) hJbj i,
2
j,b
45
where HCF (i, a) describes the crystal field environment at the (i, a)th site on the pyrochlore
lattice, discussed in Appendix A.
(0)
The mean field Hamiltonian, HMF (i, a), determines the dynamic susceptibility, χa (ω),
where a labels the sublattice site. This quantity is simply the susceptibility of the magnetic
moment Jai under the influence of only the CEF, because in the paramagnetic phase, the
second term in Eqn. (3.54) is zero. Because the interaction term is zero, we can refer to
(0)
χa (ω) as the non-interacting susceptibility, and it is given by (Eqn. 3.5.20 in Ref. [41])
Eµ 6=Eν
χ(0)
a (ω)
=
X
µ,ν
Eµ =Eν
u
v
Mνµ,a
Mµν,a
δ(ω) X
u
v
(nν − nµ ) +
Mνµ,a
Mµν,a
nν ,
Eµ − Eν − ~(ω + i0+ )
kB T µ,ν
(3.55)
P
= ū hν|Jū |µiouū,a and ouū,a is the rotation matrix from the local ū frame at
where
sublattice site a to the global u frame. The operators Jū act on the CEF states defined in
the local ū reference frames at each sublattice site. The CEF wave-functions |µi are given
in Appendix A.
u
Mνµ,a
The next step in determining the interacting RPA susceptibility is to compute the
b
b
linear response of hJai (t)i to a small perturbing field hbj (t) = gJ µB Hj (t) where Hj (t) is
an applied time dependent magnetic field. If we collect all of the terms in Eqn. (3.53) that
depend on Jai into an effective Hamiltonian Hia , we obtain (Eqn. 3.5.4 in Ref. [41])
X
a,b
Hia (t) = HMF (i, a, t) −
(Jai (t) − hJai i)T K (i, j) Jbj (t) − hJbj i + hai (t) . (3.56)
j,b
According to Ref. [41],the difference term Jbj (t) − hJbj (t)i fluctuates in an uncorrelated
manner from ion to ion, making this terms contribution to Eqn. (3.56) very small so it may
be neglected. This is equivalent to replacing Jbj (t) with hJbj (t)i for all terms where i, a 6=
j, b. This is the random
phase approximation, because it is equivalent to the assumption
b
b
that Jj (t) − hJj (t)i can be replaced by a random phase factor [41].
Introducing the RPA in Eqn. (3.56) the only dynamical variable left over is Jai (t) and
Eqn. (3.56) becomes equivalent to the mean field Hamiltonian in Eqn. (3.54), but with
hai (t) replaced by an effective field hai,eff (t). The Fourier transform of this effective field is
given by (Eqn. 3.5.5 in Ref. [41])
X a,b
hai,eff (ω) = hai (ω) +
K (i, j) hJbj (ω)i,
(3.57)
j,b
where hJbj (ω)i is the Fourier transform of Jbj (t) − hJbj i . Using these two relations, we
can write [41]
a
hJbj (ω)i = χ(0)
(3.58)
a (ω)hi,eff (ω) .
46
If we compare the form of Eqn. (3.57) to the response determined by the two-ion susceptibility function of the system defined as (Eqn. 3.5.6 in Ref. [41])
X
hJbj (ω)i =
χab (i, j, ω) hbj (ω) .
(3.59)
j,b
these two equations should be equivalent [41], which means that we can write (Eqn. 3.5.7
in Ref. [41])
!
X a,c
K (i, k) χcb (k, j, ω) .
(3.60)
χab (i, j, ω) = χ(0)
a (ω) δa,b +
k,c
a,b
Finally we can take the Fourier transform of real space interaction matrix K (i, j)
a,b
a,b
a,b
(q), where
(q) + Du,v
(q) = Ju,v
to obtain the reciprocal space interaction matrix Ku,v
a,b
Ju,v (q) is the Fourier transform of the nearest-neighbour bilinear exchange interaction
a,b
(q) is the Fourier transform
matrix, as defined in Eqn. (G.3) of Appendix G and Du,v
of the long-range dipolar interactions determined using the Ewald summation technique
discussed in Appendix E. Expressing the final result in terms of matrix elements, we obtain
the interacting RPA susceptibility χ(q, ω) (Eqn. 3.5.8 [41])
X
st
tv
0,uv
χuv
χ0,us
(3.61)
ab (q, ω) +
a (ω)Kac (q)χcb (q, ω) = δab χa (ω),
s,t,c
χuv
ab (q, ω) is then obtained by inverting Eqn. (3.61) numerically.
Now we are able compute the RPA interacting susceptibility from the CEF parameterizations of Refs. [19, 20] and our model Hamiltonian. From this we can compute the
quasi-elastic neutron scattering cross section by setting ω = 0. The next step is to extract
the form of J (i, j) from the quasi-elastic neutron scattering measurements of Chapter 1.6.
It is important to consider the validity of the use of the RPA for this calculation. For
the case of Yb2 Ti2 O7 , as discussed in Chapter 2, the gap between the ground and first
excited doublets of the CEF (∆ ∼ 650 ± 40 K) is much much larger than either the energy
scale of the of either Hex or Hdip , based on the value of θCW = 0.75 ± 0.05 K [20, 18]. As
explained in Ref. [31], this implies that there is minimal admixing between the ground and
excited CEF doublets. These calculations are also performed at a temperature of T = 1.4
K, which is approximately twice θCW for this material, and six times the experimentally
observed transition temperature of 214 mK [16]. This should place the system squarely
in the paramagnetic regime, making the RPA valid for calculations of the susceptibility.
47
3.4
Summary
In this chapter we have derived the form of the magnetic neutron scattering cross-section
from the microscopic interaction between the magnetic moment of the neutron and the
magnetic field that arrises from the electrons in a material. This derivation exposed the
fact that neutron scattering only probes correlations between the components of magnetic
moments perpendicular to the wave-vector of the scattered neutron, and that this arrises
from the form of the Fourier transformed interaction between the magnetic moment of the
neutron and the magnetic field created by the motion of the electrons in the sample. We also
examined the relationship between the quasi-elastic neutron scattering cross-section and
the wave-vector dependent susceptibility χ (q). This relationship will be very important
in Chapter 4 where we will use it in combination with the random phase approximation
method of computing χ (q) to compute the quasi-elastic neutron scattering from our model
Hamiltonian. This will allow us to compare the quasi-elastic neutron scattering crosssection generated by our model to that found in experiment in order to determine the
strengths of the symmetry allowed bilinear exchange terms in our interaction model.
48
Chapter 4
Determining the Magnetic
Interactions in Yb2Ti2O7 from
Paramagnetic Quasi-elastic Neutron
Scattering
This chapter focuses on the method used to determine the form of the magnetic interactions
in the material Yb2 Ti2 O7 from quasi-elastic neutron scattering measurements made at
T = 1.4 K on single crystal samples [31]. Specifically, we wish to determine the strengths
of the four symmetry allowed bilinear exchange terms defined in Chapter 2. The quasielastic neutron scattering data, which was discussed in Chapter 1.6 was collected by Prof.
Henrik Rønnow and Dr. Louis-Pierre Regnault. These measurements reveal the presence
of anisotropic correlations in Yb2 Ti2 O7 at T = 1.4 K via the presence of rod-like features in
the data along h111i symmetry related directions. In a completely non-interacting system
no such correlations would be present, as it is the magnetic interactions that give rise to
these correlations. Because the interactions give rise to the correlations, the form of these
correlations should be directly related to the nature of the magnetic interactions in Yb2 Ti2 O7 , and it should be possible to extract the form of the magnetic interactions in Yb2 Ti2 O7
by fitting the quasi-elastic neutron scattering. We choose to use the data collected at
T = 1.4 K rather than the data collected at T = 9.1 K as the signal to noise ratio of this
data is higher. In principal the T = 9.1 K data would have better suited our purposed,
as at this temperature Yb2 Ti2 O7 is situated much more firmly in the paramagnetic phase,
avoiding any potential problems with Yb2 Ti2 O7 being in a collective paramagnetic phase,
and correlation effects that appear due to the presence of a nearby phase transition.
49
4.1
Simulated Annealing
In this section we explain the method by which the form of the magnetic interactions is
extracted from the quasi-elastic neutron scattering of Prof. Rønnow shown in Chapter 1.6.
Initial attempts to fit the experimental neutron scattering using the model of symmetry
allowed nearest-neighbour bilinear exchange were rather haphazard, involving manual manipulation of the parameters {Jn } in an attempt to generate a neutron scattering pattern
using the RPA method similar to that seen in experiment. After many unsuccessful attempts at manual fitting, numerical optimization was decided upon as the best method
to determine if it was indeed possible to fit the experimental neutron scattering using the
proposed model Hamiltonian (Eq. (C.15)).
The numerical optimization method chosen to determine the values {Jn } is simulated
annealing. This method is closely related to the physical thermodynamic process of annealing, which is the process of slowly cooling a system in order to allow the system to
enter its absolute ground state rather than becoming trapped in a local energy minimum.
This arises in the growth of single crystal material samples. If a molten sample of some
material that forms a crystal at low temperatures is cooled rapidly, it will form a disordered
polycrystalline sample when it solidifies. On the other hand if the sample is cooled slowly
it maintains equilibrium during the process and is able to locate its lowest energy state
forming large crystals.
4.1.1
The Metropolis Algorithm
Reference [48] presents an explanation of the simulated annealing method. The basis for
simulated annealing is the Boltzmann probability distribution for a system in thermal
equilibrium [48], which is given by
P (E) ∼ exp (−E/ (kB T )) ,
(4.1)
where P (E) is the probability of a system being in a state with energy E, kB = 1.3806504×
10−23 J/K is the Boltzmann constant, and T is the temperature. This distribution expresses
the fact that even at very low temperatures there is a finite but small probability for the
system to be in a high energy state [48]. This means that there exists a corresponding
chance of escaping a local energy minimum in order to find a “better” more global minimum
via a multistep process involving a high energy state [48]. The Metropolis algorithm [49]
provides a way for this principle to be applied to numerical simulations. This algorithm is
50
based on the proposition that if the system is in a state with energy E1 , the probability of
it transitioning to a state with energy E2 is given by [48]
p = exp (− (E2 − E1 ) /kB T ) .
(4.2)
It can be seen that if E2 < E1 , this probability will be greater than 1. In this case, the
probability of the transition is simply set to 1 and the system transitions to the state with
energy E2 . In the case where the probability is less than 1, a pseudo-random number
generator (PSRNG) is used to generate a number in the range (0, 1), if the generated
number is less than p, the transition is allowed.
4.1.2
The Effective Energy Function and Implementation of the
Metropolis Algorithm
For the determination of {Jn } for Yb2 Ti2 O7 , the energy function used is in Eqn. (4.1) is
given by
X
2
2
model
− θCW ,
(4.3)
Eeff =
(c0 S(Q) + c1 + c2 |Q|) − Sexp (Q) − ∆S + Λ θCW
Q
where S(Q) is the RPA neutron scattering intensity given by Eqn. (3.49) for a specific
value of Q. Sexp (Q) is the experimental neutron scattering intensity at the same point in
reciprocal space and
∆S = mean ((c0 S(Q) + c1 + c2 |Q|) − Sexp (Q)) .
(4.4)
The scaling factors c0 , c1 , and c2 are used to relate the model neutron scattering intensity,
which is computed in arbitrary units, to the experimental neutron scattering intensity,
which is recorded in terms of number of neutrons counted. The |Q| term was found to be
important when comparing model and experimental neutron scattering cross sections in
Ref. [50]. The specific points in reciprocal space used in the sum over Q are marked by
the red lines in Fig. 4.1
model
θCW
is the Curie-Weiss temperature of the model Hamiltonian, obtained by fitting a
straight line to the inverse powder susceptibility, χ, computed at T = 2.5 K and T = 10 K,
the same temperatures used in experimental fits [20]. The value of θCW used in Eqn. (4.3)
is that of Ref. [20], θCW = 0.75 K. The powder susceptibility is given by
!
1 X uv
1
χ (q = 0, ω = 0) .
(4.5)
χ = Tr
3
4 ab ab
51
4
(h,0,0)
3
2
1
0
0
0.5
1
1.5
2
2.5
(0,k,k)
Figure 4.1: The quasi-elastic neutron scattering data at T = 1.4 K used to determine the
form of the magnetic interactions in Yb2 Ti2 O7 , with the cuts of the (h, k, k) plane used in
the simulated annealing fit marked in red.
In this equation the sum over sublattices a and b, and the trace of the resulting 3×3 matrix
represent taking the powder average of the susceptibility, consistent with what was done
in experiment. The sum over a and b of χuv
ab (q = 0, ω = 0) yields 4 times the single crystal
susceptibility, which is diagonal. Taking the trace of this result and dividing by 3 results
in the average of the susceptibility in all of the global cartesian coordinates, which is what
is recorded in powder susceptibility measurements because of the random orientations of
the powder grains.The value of Λ was chosen ad hoc to be 100000, in order to give this
term the same weight in the calculation of Eeff as the neutron scattering term.
The following procedure is used to determine the values of {Jn } and {cn } = {c0 , c1 , c2 }
that, when used in the model of Eqn. (C.15) to compute the RPA neutron scattering, result
in the best fit to the experimental quasi-elastic neutron scattering data. First, a random
initial set of interaction energies {Jn } and scaling parameters {cn } are chosen, along with
a beginning effective temperature Teff to start the process. We then proceed as follows:
1. Choose a parameter P from {Jn } or {cn } at random using a PSRNG.
2. Using a PSRNG once again, generate a random value in the range (−1, 1). Multiply
this value by the maximum allowed step size for P and add this value to P to form
Pnew .
52
init
fin
3. Compute Eeff using P and Pnew , yielding the values Eeff
and Eeff
.
init
fin
fin
init
/Teff .
− Eeff
, compute p = exp Eeff
and Eeff
4. Using Eeff
5. If p > 1 set P = Pnew . If not, generate a random number z in the range (0, 1), if
z < p, set P = Pnew , otherwise do nothing.
6. Return to step 1.
This process is repeated 4000 times for a given Teff , at which point Teff is lowered to
Teff / (1 + κ), a process that is performed 100 times, leading to 400000 Monte Carlo steps
in total. This part of the process is known as the annealing schedule [48].
This process will not yield the absolute minimum Eeff , and thus the optimal values of
{Jn } and {cn }, using one single annealing run, even if massive numbers of Monte Carlo
steps and a very slow annealing schedule are used. To account for this, the code was run
100 times in parallel using different random seeds for the PSRNG, to obtain a distribution
of results. The results of this process are discussed in the next section.
4.1.3
The Curie-Weiss Temperature Constraint and Additional
Refinement of Simulated Annealing Results
As discussed in Chapter 4.1.2, the second term of Eqn. (4.3) is proportional to the difference between the experimental Curie-Weiss temperature for Yb2 Ti2 O7 , θCW = 0.75 K [20],
model
. This constraint is inand the Curie-Weiss temperature of the model Hamiltonian, θCW
cluded in order to impose the correct energy scale on the interactions determined using the
simulated annealing fit to the experimental quasi-elastic neutron scattering data, because
this energy scale is directly related to θCW .
θCW in Ref. [20] is determined by fitting a straight line to experimental measurements
of the inverse powder magnetic susceptibility at temperatures between T = 2.5 K and
model
T = 10 K. The temperature axis intercept of this line is θCW . θCW
is determined by
fitting a straight line to the inverse susceptibility calculated at T = 2.5 K and T = 10 K.
model
This simple fit was chosen in order to make the process of determining θCW
simple and
computationally efficient in order to make the calculations run quickly. If computational
model
efficiency had not been an issue, a more ideal way to compute θCW
would have been to
−1
compute χmod , the inverse susceptibility of our magnetic interaction model at the same
points at which χ−1
exp , the experimental inverse magnetic susceptibility, was measured.
53
If this method had been used, then we could have replaced the soft
2 constraint imposed
model
model
on θCW via the second term in Eqn. (4.3), E2 = Λ θCW − θCW , with a functionally
equivalent term
X
2
−1
E20 = Λ
χ−1
.
(4.6)
exp,i − χmod,i
i
In this new new term, χexp,i and χmod,i are the experimental and model values of the
powder magnetic susceptibility at specific temperature points indexed by the label i. The
sum over i indicates the sum over all of the temperatures at which the experimental powder
susceptibility is available. As the number of points i in E20 increases, P (Eeff ), which contains
the terms exp (−E20 /T ) will look more and more like a delta function,
−1
δ {χ−1
exp,i }, {χmod,i }
−1
, where {χ−1
exp,i } and {χmod,i } are the sets of all the points for which χexp and χmod are
model
available. This delta function is equivalent to the delta function δ θCW , θCW
, because
the inverse powder susceptibility and θCW are directly related. As the fit between {χ−1
exp,i }
−1
and {χmod,i } is further and further constrained by the increasing number of points, so is
model
.
the fit between θCW and θCW
This discussion of using a constraint based directly on the inverse powder susceptibility
rather than on θCW is directly related to the issue of filtering the results of our simulated
annealing method. This filtering is required because the simulated annealing fit process
will result in a large cloud of points in {Jn } space, not all of which are actually models that
fit the experimental neutron scattering pattern or θCW . This poor fitting occurs because
model
the constraints on θCW
and S(Q) are soft constraints, meaning that these values are
allowed to vary in an attempt to minimize the global function Eeff , and not fixed to the
precise values found in experiment by a delta-function like constraint. Fig. 4.2 shows S(Q)
for two sets of couplings, {Jn }, the results of two different simulated annealing runs. One
of these sets of {Jn }, matches S(Q)exp and θCW = 0.75 K [20] well, while the other fits
model
them poorly, with θCW
= 0.78 K. Such poorly fitting {Jn } were eliminated from further
model
consideraction, with any result where θCW
varied from 0.75 K by more than 0.01 K being
discarded. This stringent constraint is justified by the previously explained argument,
whereby an improved method of fitting θCW , based on a fit to {χ−1
exp,i }, would result in a
model
very stringent constraint on θCW .
54
Experiment
=0.75 K
=0.78 K
4000
3800
3600
3400
3200
3000
2800
2600
2400
2200
2000
0
50
100
150
200
250
300
350
400
450
Figure 4.2: S(Q) plotted for the sets of couplings, {Jn }, resulting from two different
model
= 0.78 K, while the other yields
simulated annealing runs. One set of {Jn } yields θCW
model
model
= 0.75
θCW = 0.75 K. It can be seen quite clearly that S(Q) for the set of {Jn } with θCW
model
K fits the experimental data much better than the set of {Jn } with θCW = 0.78 K. Qi
denotes all of the points in the sum over Q in Eqn. (4.3) appended into a single set, as
they would be stored in our simulated annealing program.
4.2
Results
We now present the results of simulated annealing fits to the quasi-elastic neutron scattering
data collected by Prof. Rønnow [31], discussed in Chapter 1.6. Simulated annealing fits
were performed using two slightly different models. These two models are differentiated
by the CEF parameterization used to compute the single ion susceptibility in Eqn. (3.55).
Each of the two sets of simulated annealing fits resulted in a cloud of points in {Jn }
space. As previously discussed in Chapter 4.1.3, some of these points are not in fact good
fits to the experimental neutron scattering data and/or the experimental value of θCW
used in the optimization, θCW = 0.75 K [20], due to the model becoming trapped in a local
minimum of Eeff . These poorly fitting points were discarded for both fits with any result
where θCW model varied from 0.75 K by more than 0.01 K being discarded. The resulting
points are displayed in {Jn } space for both the CEF of Hodges et al. [20] and Cao et al.
55
[19] in Fig. 4.3. As can be seen, both CEF parameterizations result in tight clusters in {Jn }
space. The average values of these clusters are marked by the crossing points of the three
red lines (in J1 , J2 , J3 ). The exact values of these points for both CEF parameterization
used are reported in Table 4.1 along with uncertainties of 1 standard deviation.
Table 4.1: Average values of the coupling parameters {Jn } resulting from simulated annealing. The indicated uncertainties correspond to one standard deviation.
CEF Parameterization
J1
J2
J3
J4
Hodges et al. [20]
0.100 ± 0.005 K 0.202 ± 0.003 K 0.166 ± 0.006 K 0.001 ± 0.004 K
Cao et al. [19]
0.061 ± 0.003 K 0.171 ± 0.002 K 0.139 ± 0.004 K 0.001 ± 0.003 K
Using the relationship between {Je } = {JIsing , Jiso , Jpd , JDM }, and{Jn }, discussed in
Chapter 2.2.2 and Appendix C.2, we can rewrite the coupling values of Table 4.1 in terms
of {Je }, obtaining the results in Table 4.2. This notation will be used from now on.
If we examine the coupling values of {Je } we can see that the largest term in both sets
of {Je } is the local Ising term JIsing , which is very interesting as it is in direct opposition
to the planar nature of the ground state doublet of the both CEF parameterizations. The
other three couplings are similar in magnitude, being approximately one quarter of the
value of JIsing . It is interesting to note that the negative sign of Jpd in Tables 4.2 and
4.3, would correspond to physical, as opposed to unphysical, dipolar interactions, if this
interaction were the long-range magnetic dipolar interaction. Finally it is interesting that
JDM is on the order of the other couplings. The DM interaction arises as a perturbative
term due to spin orbit coupling in magnetic materials [51] and because it arrises due to
perturbation theory, it is expected to be quite small. Recent work by Onoda et al. [52, 53]
has shown that is indeed possible for values of the DM coupling on the order of those found
here to exist in the case of Yb2 Ti2 O7 . This work is discussed further in Appendix H.
The black dots in Fig. 4.3 mark chosen values for direct comparison of the model results
to experiment, and whose specific values of {Je } are given in Table 4.3. Exact rather than
average values are used as precise scaling parameters {cn } are required for comparison
between model and experiment (reported in Table 4.4), and averaging these would lead to
a poor fit. The scattering in the (h, k, k) plane generated by the two models in Table 4.3 are
shown in Fig. 4.4(a,b). Examining the RPA scattering for both models it can be seen that
they successfully reproduce most of the features of interest in the experimental data shown
in Fig. 4.4(d), including the [1, 1, 1] rod of scattering, the rod-like feature between (2, 2, 2)
56
(a)
(b)
(c)
(d)
Figure 4.3: The refined results of simulated annealing optimization for the CEF parameterizations of (a,c) Hodges et al. [20] and (b,d) Cao et al. [19]. It can be seen in (a) and
(b) that the refined clusters are quite small in {Jn } space. The red lines illustrate the
location of the average value of the cluster, given in Table 4.1. Panels (c) and (d) show
closer views of the clusters. The red lines once again mark the average value of {Jn } for
the cluster, with the small black dots showing one standard deviation in each of J1 ,J2 ,
and J3 . J4 is omitted from this treatment as it is shown on the colour scale and its range
is small. The large black dots in panels (c) and (d), highlighted by arrows, indicate the
simulated annealing results that have been selected for direct comparison to experiment.
57
Table 4.2: Average values of the coupling parameters {Je } computed from the values of
{Jn } in Table 4.1.
JIsing
Jiso
Jpd
JDM
Hodges et al. [20]
0.80 ± 0.02 K 0.223 ± 0.003 K −0.289 ± 0.009 K −0.267 ± 0.009 K
Cao et al. [19]
0.75 ± 0.02 K 0.181 ± 0.002 K −0.259 ± 0.006 K −0.248 ± 0.006 K
and (0, 0, 4), and the intensity around the point (2, 2, 0). It is also interesting to note that
neither model reproduces the feature centred around (2.25, 2.24, 3.5). This means that the
model has successfully postdicted the fact that this feature is not magnetic in origin, as
discussed in Chapter 1.6. Figure 4.4(c) shows the result of attempting the reproduce the
experimental data using only isotropic exchange (JIsing = Jpd = JDM = 0), with strength
Jiso = 0.06 K, determined from a single parameter fit to match θCW = 0.75 K [20, 31]. It
can be seen quite clearly that this model fails to reproduce any of the features of interest
in the experimental data.
Table 4.3: Selected values of {Je } for comparison of model neutron scattering to experiment. These sets of {Je } are taken from near the centre of the cluster of the results of
simulated annealing optimization shown in Fig. 4.3. θCW for the models, computed using
the method given in Chapter 4.1 is also shown.
JIsing
Jiso
Jpd
JDM
θCWmodel
Hodges et al. [20]
0.808767 K 0.0.223796 K −0.289994 K −0.267714 K 0.750093 K
Cao et al. [19]
0.755082 K 0.180150 K −0.258359 K −0.247568 K 0.753445 K
Figure 4.5 shows a quantitative comparison of experimental and model neutron scattering data at T = 1.4 K for both models in Table 4.3 and the simple isotropic exchange
model. This comparison allows us to see that both models are not only excellent qualitative fits to the experimental neutron scattering pattern, as seen in Fig. 4.4, but are also
excellent quantitative fits to experimental neutron scattering intensities. This is in contrast
to the isotropic exchange model, which is confirmed to be a very poor fit to experiment
despite separate annealing to determine the best possible set of coupling parameters {cn }
for this model. This poor fit occurs despite this separate annealing because the neutron
scattering pattern of the isotropic exchange model is so fundamentally different from that
58
(a)
(b)
4
(h,0,0)
3
2
1
0
0
(c)
(d)
0.5
1
1.5
2
2.5
(0,k,k)
Figure 4.4: Quasi-elastic neutron scattering computed at T = 1.4 K for the models in
Table 4.3. Panel (a) shows the neutron scattering for the coupling parameters obtained
using the CEF of Hodges et al. [20]. Panel (b) shows the neutron scattering for the
coupling parameters obtained using the CEF of Cao et al. [19]. Panel (c) shows the
neutron scattering generated by a model using only isotropic exchange (Jiso = 0.06 K) and
long-range dipolar interactions. Panel (d) shows the experimentally obtained quasi-elastic
neutron scattering at T = 1.4 K of Ref. [31] for comparison.
59
of experiment that even the optimal combination of {cn } that minimizes Eeff results in a
poor fit to experiment. Fig. 4.5a does appear to show a good fit between the neutron scattering of the isotropic exchange model and experiment. This occurs because the cut along
(h, 0.05, 0.05) in reciprocal space passes through a region of the neutron scattering pattern
where there are no magnetic features, and only background noise, in both experiment and
the scattering of the isotropic exchange model. This means that the data in Fig. 4.5(a)a
does not actually represent any kind of agreement between the isotropic exchange model
and experiment, it only shows that we can match the background noise level.
Table 4.4: Values of the scaling parameters {cn } for the two sets of couplings {Je } reported
in Table 4.3 that are required for quantitative comparison of RPA neutron scattering
calculations to experiment.
Hodges et al. [20]
Cao et al. [19]
c0
38.625395
39.181870
c1
1700
1700
c2
13.406493
20.618574
The result of all of these comparisons is that we have two potential models, one for
each of the CEF parameterizations used that are capable of explaining the features seen
in the experimental paramagnetic quasi-elastic neutron scattering data.
Now we must place some constraints on our results. These constraints arise from the
use of the RPA, specifically the assumptions about the energy resolution. More rigorous
calculations of dσ/dΩ(q) using χ (q, ω) and then integrating over a finite range of dω were
performed by Dr. Paul McClarty. These calculations [31] found very similar, but not quantitatively identical, scattering patterns in the (h, k, k) plane to those in Fig. 4.4(a,b). This
indicates that some small changes in {Je } would be evident if the simulated annealing fit
were to be performed using exact integration over the same energy range measured by the
detectors in the experiment. In addition to this, when using the RPA to perform calculations it is important to consider the possible presence of correlations due to the proximity
of a phase transition and not just magnetic interactions. Our calculations were performed
at T = 1.4 K, which as previously discussed, and noted in Ref. [31], is approximately five to
six times the reported temperature of the phase transition in Yb2 Ti2 O7 , Tc ≈ 214 mK [16],
and twice the experimentally reported value of θCW = 0.75 ± 0.05 K [18, 20]. Given this
proximity to the observed phase transition some correlation effects, due to the divergence
of the correlation length near the phase transition, are to be expected. If the effects of these
correlations related to the phase transition were to be accounted for, some change in the
60
4000
(a)
5500
(b)
4000
3500
3000
3000
Intensity
4500
3000
2500
0
1
2
3
(h,0.05,0.05)
2000
0
4
1
2
(h,0.5,0.5)
3
(e)
3600
Intensity
Intensity
Intensity
4000
3200
3000
2800
2200
1
2
3
(h,0.95,0.95)
3600
0
4
(g)
1
2
3
(h)
3000
3000
2800
Intensity
3200
Intensity
Intensity
(f)
2800
2600
2600
2400
1
2
3
(h,1.75,1.75)
4
1
2
3
(h,2.25,2.25)
(k)
(h,0,0)
3
3200
3000
2800
1
2400
1.5
(4−k,k,k)
2
2.5
0
0
(i)
2600
bcd e f g
0.5
1
1.5
(0,k,k)
h
2
1
1.5
(3−k,k,k)
2
2.5
experiment
model #2
i
a
0.5
model #1
j
2
2600
4
2800
0
4
4
3400
3
2200
0
(j)
3600
2
(h,1.5,1.5)
2400
2200
2400
1
3000
2800
2600
Intensity
4
3000
2200
0
4
(h,1.25,1.25)
3400
1
3
2400
2400
2200
0
2
(h,0.75,0.75)
2600
3000
2000
0
1
3200
3400
5000
2600
2200
0
4
3800
(d)
6000
2800
2400
2500
2000
(c)
3200
3500
Intensity
Intensity
5000
2.5
isotropic &
dipoles
Figure 4.5: Panels (a) through (j) show cuts through the (h, k, k) plane in reciprocal space. The
blue dots show the experimental paramagnetic quasi-elastic neutron scattering at T = 1.4 K.
The solid red lines show the results of RPA calculations of the quasi-elastic neutron scattering at
T = 1.4 K using the set of couplings {Je } shown in Table 4.3 for the CEF parameterization of
Hodges et al. [20] (model #1). The solid green lines show the same calculation, but using {Je }
obtain using the CEF parameterization of Cao et al. [19] (model #2). The black dashed line
shows the result of RPA calculations using only isotropic exchange (Jiso = 0.06 K) discussed in
the text. Panel (k) shows a map of the (h, k, k) plane, indicating the positions of all of the cuts
in panels (a) through (j).
61
values of {Je } determined using the current method would be expected, possibly on the
order of 1/ (T − θCW )2 ∼ 25%, the leading correction to the high temperature expansion
of χ(q) [31]. Despite these constraints, the RPA calculation of dσ/dΩ(q) will generate a
unique scattering pattern in the (h, k, k) plane for a given set of couplings {Je }, so the
variation of the couplings when these constraints are taken into account should be on the
order of one [31]. The overall energy scale of the couplings {Je } also cannot change drastimodel
cally due to the imposition during the simulated annealing fit that θCW
must match that
found in experiment, placing a constraint on the energy scales of the interactions. The net
result of these constraints and the assumption of bilinear exchange discussed earlier is that
the Hamiltonian we find to describe the magnetic interactions in Yb2 Ti2 O7 will capture
the important elements of the low temperature physics of Yb2 Ti2 O7 . It will not capture
the physics of Yb2 Ti2 O7 at large energies, on the order of the CEF splitting.
RPA calculations can also provide us with other interesting properties of our model
Hamiltonian; those being the RPA transition temperature TRPA and the ordering wavevector. TRPA is determined by lowering the temperature at which the neutron scattering
is computed until the scattering cross section becomes negative for some point in q space.
The temperature at which this occurs will be directly proportional to the largest eigenvalue
of J (q) anywhere inside the first Brillouin zone. The mean field ordering wave-vector is
the point q at which the largest eigenvalue occurs 2 . By searching over the first Brillouin
zone, we find that both models yield q = 0 order, with values of TRPA ∼ 1.17 K.
2
The mean field free energy of a system, F ({mai }), is a function of {mai }, the set of all of the vector
order parameters mai for all of the sites on the pyrochlore lattice [54, 55]. If we then Fourier transform
the free energy into momentum space, it takes the from F ({ma (q)}), where {ma (q)} are the Fourier
transformed order parameters. Then, in order to diagonalize the Fourier transformed interaction matrix
ab
K (q), {ma (q)} must be transformed into normal mode variables, whose amplitude are denoted φα
q µ,
where α and µ label the normal modes [54]. The quadratic term in
the
free
energy
expressed
in
terms
of
α 2
P
these normal mode amplitudes (Eqn. 10 in Ref. [54]) is given by 21 q,α,µ nT − λα
µ (q) |φq µ| , where n is
the dimension of the spin (n = 3 for Heisenberg spins), T is the temperature, and λα
µ (q) are the eigenvalues
of the Fourier transformed interaction matrix. From this result it can be seen that if the coefficient of one of
the normal mode amplitudes becomes negative, the value of the normal mode amplitude which minimizes
F ({ma (q)}) will become non-zero, and the system will order. The highest temperature at which the
coefficient for one of the normal mode amplitudes becomes negative will correspond to the largest λα
µ (q)
inside the first Brillouin zone, and the mean field ordering wave-vector is the wave-vector q at which this
eigenvalue achieves its maximum value.
62
4.3
Summary
In this chapter we have used fits to experimental quasi-elastic neutron scattering patterns to determine the strength of the four symmetry allowed bilinear exchange terms in
our model of the magnetic interactions inYb2 Ti2 O7 . Quasi-elastic neutron scattering is
used to determine the form of the interactions because it is directly related to the interaction driven correlations between the magnetic moments in Yb2 Ti2 O7 . In order to
compute the quasi-elastic neutron scattering from our proposed model, we use the random phase approximation (RPA) and compared RPA neutron scattering and experiment
using a simulated annealing process to determine optimal strengths of the various bilinear
exchange terms. The results of this simulated annealing show that a model containing
anisotropic nearest-neighbour bilinear exchange and long-range magnetic dipolar interactions can successfully reproduce the important features seen in the experimental quasielastic neutron scattering. The model that we obtain consists of nearest-neighbour local
Ising exchange, nearest-neighbour isotropic exchange, nearest-neighbour pseudo-dipolar
exchange, and Dzyaloshinskii-Moriya exchange. We find that nearest-neighbour local Ising
exchange is the strongest term in our model for Yb2 Ti2 O7 , being approximately four times
the strength of the other three exchange terms. We also find that Dzyaloshinskii-Moriya
exchange is abnormally large in our model.
63
Chapter 5
Real Space Correlations
In Ref. [40] it was proposed that the rod-like features in the paramagnetic quasi-elastic
neutron scattering discussed in the previous chapter arise from two-dimensional correlations
within the kagome planes that make up the pyrochlore lattice. This is because neutron
scattering is a probe magnetic correlations between the elements of the magnetic moments
J perpendicular to the scattering wave-vector Q. If we have a system with magnetic
correlations in some direction r̂, with characteristic length γ, then the neutron scattering
cross section in a direction Q̂ k r̂ will exhibit a peak around Q = 2π/γ, as shown in
Fig. 5.1(a). If there are no correlations along a given Q, the neutron scattering along
that direction in reciprocal space will consist only of background scattering, as shown in
Fig. 5.1(b). If correlations exist in two dimensions, but not in a third, these two types of
scattering will be superimposed, as shown in Fig. 5.1, resulting in a rod of scattering in
reciprocal space.
Now that we have determined two sets of magnetic interactions for Yb2 Ti2 O7 that
successfully reproduce the neutron scattering pattern seen in experiment, we can test
whether this pattern does in fact arise from two-dimensional correlations as suggested
in Ref. [40], by actually computing the spin-spin correlation function along several high
symmetry directions of the pyrochlore lattice within the random phase approximation
(RPA).
64
(a)
(b)
dσ
dΩ
dσ
dΩ
2π/ɣ
Q
Q
(c)
(d)
Q
Q
z
z
Qy
Qy
Qx
Qx
Figure 5.1: Panel (a) shows the neutron scattering intensity along Q in reciprocal space,
arising from real space magnetic correlations along r̂ k Q̂, with characteristic length γ.
Panel (b) shows the neutron scattering intensity when there are no such correlations.
Panel (c) shows the rod of neutron scattering intensity in 3D reciprocal space that arises
from the superposition of correlations in two directions, i.e. two-dimensional correlations.
Panel (d) shows the plane of neutron scattering intensity in 3D reciprocal space that arises
from correlations in one direction in reciprocal space, with the black arrows showing the
directions in which the plane extends.
65
5.1
Computing the Real Space Correlation Function
in the Random Phase Approximation
In order to compute the real space correlations, we return to the RPA. We saw in Chapter 3
that there is a direct relationship between the wave-vector dependent susceptibility and the
correlation function hJia,u Jjb,v i. This means that it is possible to extract the real space correlation function hJia,u Jjb,v i from the RPA susceptibility. The specific relationship required
for this calculation can be shown in the following manner. We start from Eqn. (3.43)
1
uv
uv
(Q, 0) − Sab,Bragg
(Q, 0)
Sab
kB T
!
X
X
exp iQ · Rai − Rbj hJia,u Jjb,v i −
exp iQ · Rai − Rbj hJia,u ihJjb,v i .
χu,v
a,b (Q, 0) ∝
∝
1
kB T
i,j
i,j
By taking the inverse Fourier transform of this equation, we obtain the result
Z
a,u b,v
hJi Jj i ∝
exp −iQ · Rai − Rbj χu,v
a,b (Q, 0) dQ,
(5.1)
BZ
where the integral over dQ is taken over the first Brillouin zone of the FCC lattice. If we
take one of the moments to be at the origin, by setting i = 0, a = 1, we can vary j and
b to determine the Rab
ij dependence of the real space correlations along different directions
in real space. This integral is performed using Monte Carlo integration. We choose to
compute two quantities using this method:
Z
exp −iQ · Rai − Rbj χu,u
S(r) =
(5.2)
a,b (Q, 0) dQ,
BZ
which is proportional to hJai · Jbj i, and
Z
S⊥ (r) =
exp −iQ · Rai − Rbj
δu,v − Q̂u Q̂v χu,v
a,b (Q, 0) dQ,
(5.3)
BZ
which is directly related to the inverse Fourier transform of the neutron scattering cross
section. Now that we can compute the real space correlation functions from the wave-vector
dependent susceptibility, it is a simple matter to use the RPA method of Chapter 3.3 to
compute the wave-vector dependent susceptibility in the RPA and use this to compute the
real space magnetic correlation functions.
66
5.2
Results
Using the set of anisotropic exchange couplings for the CEF parameterization of Hodges
et al. from Table 4.3, we have computed the real space correlation functions S(r) and
S⊥ (r) along three high symmetry directions of the pyrochlore lattice. We feel justified in
computing these correlations using only one of the two models obtained in Chapter 3 as
the scattering patterns generated by the two models are almost indistinguishable. As the
scattering cross section is directly related to the magnetic correlations this means that the
real space correlations generated by the two models should also be essentially indistinguishable. The three high symmetry directions are [01̄1], a nearest-neighbour chain direction,
[12̄1], a second nearest-neighbour direction, and [111], a cubic unit cell body diagonal direction, perpendicular to one of the kagome planes in the pyrochlore lattice. As our model
does not break any of the symmetries of the pyrochlore lattice, all symmetry equivalent
nearest-neighbour chain directions, second nearest-neighbour directions, and h111i cubic
body diagonal directions will display identical magnetic correlations. This means that we
need only compute the real space correlations along one example of each of these high symmetry directions. This symmetry of the correlation function was checked by computing
the real space correlation functions along various symmetry equivalent directions on the
pyrochlore lattice, and it was found that the correlation functions behave identically along
symmetry related directions. These high symmetry directions are shown in Fig. 5.2.
The results of our calculations, performed at T = 1.4 K are shown in Fig. 5.3. There,
it can be seen that both correlations functions, S(r) and S⊥ (r), behave in similar manners as a function of distance. As the distance from the origin increases, the correlations
functions decay in an exponential-like manner with the three directions splitting into two
groups in terms of the strengths of the correlations. The strongest correlations are found
along the nearest-neighbour chain direction [01̄1], for both S(r) and S⊥ (r). Perhaps most
interestingly, the correlation functions S(r) and S⊥ (r) have very similar values along the
second nearest-neighbour direction [12̄1] and cubic body diagonal direction [111].
In the case of two dimensional correlations within the kagome planes, we would naively
expect a three way, rather than two way splitting of the correlation functions S(r) and
S⊥ (r). This is based on the fact that any nearest-neighbour chain within the pyrochlore
lies in two of the kagome planes that make up the pyrochlore lattice, as illustrated in
Fig. 5.2. The second nearest-neighbour directions of the pyrochlore lattice lie within only
one of these kagome planes, and the cubic unit cell body diagonals are not parallel to any
kagome planes. The fact that the nearest-neighbour directions lie in two kagome planes
means that if we had correlations within the kagome planes, but not between them, the
correlations along nearest-neighbour chains would have contributions from the correlations
67
Figure 5.2: High symmetry directions of the pyrochlore lattice. [01̄1] (red) is one of the
nearest-neighbour chain directions of the pyrochlore lattice. [12̄1] (green) is one of the
second nearest-neighbour directions of the pyrochlore lattice that lies within the horizontal
kagome plane shown. [111] (blue) is one of the cubic unit cell body diagonal directions
of the pyrochlore lattice. Two of the kagome planes that can be used to construct the
pyrochlore lattice are shown.
within two kagome planes. This would make these correlations stronger than the correlations along the second nearest-neighbour directions, which would have contributions from
only one kagome plane. In addition the correlations along the second nearest-neighbour
directions would be stronger than the correlations along the cubic unit cell body diagonals because they are parallel to no kagome planes. This is not consistent with the
two-way splitting we observe in S(r) and S⊥ (r) computed from our model. We therefore
conclude that two dimensional correlations within the kagome planes in the pyrochlore
lattice are not the source of the rods of neutron scattering intensity in Yb2 Ti2 O7 . Instead,
if we return to Fig. 5.3 and examine the correlation function S(r), we can see that the
strongest correlations are along the nearest-neighbour chains. Based on this observation
we conclude that the rods of scattering somehow arise from the superposition of strong
correlations along all of the nearest-neighbour chains that make up the pyrochlore lattice.
If the correlations were only along parallel chains, Yb2 Ti2 O7 would exhibit planes of scattering, as shown in Fig. 5.1(d), resulting from a superposition of the scattering resulting
from correlations in one direction, and no correlations in two others. However, according
to our model the correlations along all of the nearest-neighbour chains are the same. If
these chains were orthogonal, it would lead to “spots” in the neutron scattering intensity,
68
arising from the superposition of correlations in three dimensions. Instead the pyrochlore
lattice geometry somehow results in the superposition of nearest-neighbour chain correlations leading to rods of neutron scattering intensity. This model of isotropic correlations
along all nearest-neighbour chains within the pyrochlore lattice should be equivalent to
isotropic correlations between the spins, as would be found for a Heisenberg ferromagnet
on the pyrochlore lattice. Interestingly, preliminary investigations of the Heisenberg ferromagnet on the pyrochlore lattice, discussed in Chapter 7, also find rod-like features in the
paramagnetic quasi-elastic neutron scattering.
We have also computed the correlation lengths of S(r) along the three high symmetry
directions of the pyrochlore lattice by fitting the correlations in Fig. 5.3(a) to the function
a exp (−|r|/γc ). Along the nearest-neighbour chain direction [01̄1], the correlation length is
γc = 3.8 ± 0.3 Å. Along the second nearest-neighbour direction [12̄1], the correlation length
is γc = 3.4 ± 0.6 Å, and along the cubic unit cell body diagonal the correlation length is
γc = 3.2 ± 0.5 Å. These correlation lengths are not significantly different from one another,
which is strange given that the correlation function magnitudes show such clear separation
between nearest-neighbour chain directions, and the other high symmetry directions.
5
5
[01̄1]
0
0
[12̄1]
−5
−5
−10
−10
[111]
a
−15
0
5
10
b
−15
0
5
10
Figure 5.3: Figure reproduced from Ref. [31]. Real space correlations functions S(r) (a)
and S⊥ (r) (b), computed at T = 1.4 K, plotted along three high symmetry directions of
the pyrochlore lattice.
One other possible method to examine the real space correlations is to look at neutron
scattering in planes perpendicular to the [111] rod, as shown in Fig. 5.4. The scattering
in these planes is computed using the same model Hamiltonian as used to compute the
69
real space correlation functions, described by the set of couplings {Je } from Table 4.3 for
the CEF parameterization of Hodges et al. [20]. In Fig. 5.4 we can see that the rods
have a distorted triangular cross section, being widest along the [12̄1] symmetry related
directions, due to the presence of the other symmetry related h111i rods and other features
in the scattering. This cross section is consistent with our calculations of the real space
correlations. The width of the rod in a given direction in reciprocal space is inversely
proportional to the correlations in that direction in real space. This means that if the rod
is wide in a certain direction, it is expected that the correlations in that direction will fall off
more quickly than in directions where the rod is narrower. As we can see in Fig. 5.4, the
rod is wider along the [12̄1] symmetry related second nearest-neighbour directions than
along the [01̄1] symmetry related nearest-neighbour chain directions. This is consistent
with the correlations along the [12̄1] symmetry directions being weaker than those along
the [01̄1] symmetry related directions. Based on this logic, if the rods of scattering intensity
were indeed due to isotropic correlations in the kagome planes, we would expect the rod
to be perfectly circular in cross section. Therefore these neutron scattering plots provide
further evidence that correlations within but not between kagome planes is not the correct
explanation of the rods.
5.3
Summary
To summarize, we have calculated the real space correlations that result from our model
of anisotropic nearest-neighbour bilinear exchange along high symmetry directions of the
pyrochlore lattice. Ref. [40] suggested that the rods of scattering intensity in Yb2 Ti2 O7
arise due to correlations within, but not between, the kagome planes that make up the
pyrochlore lattice. The results of our calculations are not consistent with this picture.
Instead we find that the rods arise due to strong correlations along all of the nearestneighbour chains in the pyrochlore lattice.
70
Figure 5.4: RPA Quasi-elastic neutron scattering computed in three planes perpendicular
to the direction [111] in reciprocal space, passing through the points (0.5, 0.5, 0.5),(1, 1, 1),
and (1.5, 1.5, 1.5). The high symmetry directions [01̄1] (red), [12̄1] (green), and [111] (blue)
are shown.
71
Chapter 6
The Local Susceptibility
Now that we possess a pair of Hamiltonians that can successfully reproduce the paramagnetic quasi-elastic neutron scattering of Yb2 Ti2 O7 , we wish to obtain an a further
confirmation that our models correctly describe the magnetic interactions in Yb2 Ti2 O7 .
In order to do this, we require an experimentally measurable property of Yb2 Ti2 O7 that
depends on the magnetic interactions between Yb3+ ions and that we can compute using
our model. The quantity that we choose for comparison between experiment and model
results is the local susceptibility, χa . The local susceptibility was discussed briefly in Chapter 1, where the recent works in which measurements of this quantity were reported and
discussed, Refs. [19, 34, 37], were summarized. In this chapter we discuss what χa is, and
how it is measured in experiment. We will also present calculations of χa based on the
anisotropic exchange Hamiltonian found for Yb2 Ti2 O7 in Chapter 3.
6.1
Definition of the Local Susceptibility
, the change of the of
In magnetic materials the bulk susceptibility χ is defined as χ = ∂M
∂H
the system magnetization M with respect to an external magnetic field H. In a full three
u
, where u and v
dimensional system, χ is actually a 3 × 3 tensor, with elements χuv = ∂M
∂H v
index the global cartesian components of M and H, respectively the vector magnetization
and external magnetic field. The bulk magnetization describes the response of the average
moment of a magnetic system to an external field. The average moment is the sum of
all individual magnetic moments divided by the total number of moments. The local
susceptibility can also be defined. It describes the response of a single magnetic moment,
Ma , to the external field, via the equation Ma = χa H, where χa is χa expressed in the
72
global coordinates. The relationship between χa and χa is given by χa = ouū,a χa ouū,a T , where
ouū,a is the transformation matrix from the local coordinate system at sublattice site a given
in Table 1.3 to global cartesian coordinates.
The relationship between the bulk and local
P
1
susceptibility is given by χ ≡ Ns a χa , where the sum is over all sublattice sites, and Ns
is the number of sublattice sites. If χa is the same for all sites on the lattice, then this
relationship simplifies to χ ≡ χa . If on the other hand χa is not the same for all sublattice
sites, the sum over the sublattice sites can lead to the bulk and local susceptibilities having
very different properties.
In the case of the rare earth pyrochlore oxides, particularly Yb2 Ti2 O7 , the crystal structure is formed of a tetrahedral sublattice structure on an FCC lattice. At each sublattice
site, the CEF imposes a different asymmetry on the magnetic moments, leading to an
inequivalence of χ and χa . In the case of Yb2 Ti2 O7 , the magnetic moments prefer to lie
in easy planes (see Fig. 1.4). This means that the components of χa will be greatest in
these planes, but the sum over all of the sublattices will be completely isotropic, and thus
measurements of the bulk susceptibility cannot access any of the easy plane physics in this
material. If we define χa in terms of the local coordinate systems defined in Table 1.3,
then in zero magnetic field, χa has two components, χk and χ⊥ , and is of the form

χ⊥ 0 0


χa =  0 χ⊥ 0  .
0
0 χk

(6.1)
In global coordinates, χa is different at each sublattice site. If we choose sublattice one,
then it has the form


χ1,1 χ1,2 χ1,3


χa =  χ2,1 χ2,2 χ2,3  .
(6.2)
χ3,1 χ3,2 χ3,3
In the case of local easy planar symmetry χ1,1 = χ2,2 = χ3,3 , and χ1,2 = χ1,3 = χ2,1 =
χ2,3 = χ3,1 = χ3,2 . The relationship between the components of the local susceptibility in
these two representations is χ⊥ = χ1,1 − χ1,2 and χk = χ1,1 + 2χ1,2 . These four relationships
can be found quite simply by, applying the rotation matrices ouū,a to Eqn. (6.2).
Now that we have defined the quantity we wish to compute, we can look at how it is
measured in real materials. This is done by using the results of polarized neutron scattering
measurements, following the method of Gukasov and Brown laid out in Ref. [35]. As already
stated, χa is the quantity relating the magnetic moment at a single sublattice site to the
external field, that is Ma = χa H. The same holds for all sublattice sites, meaning that
73
for another sublattice site, b, Mb = χb H. The local susceptibility at these two sublattice
sites are, following Ref. [35], related by the equation χb = R̃χa R̃−1 . R̃ is the rotation
matrix component of the space group symmetry operation connecting the two sublattice
sites, {R̃ : t}, and t is the translation vector between the sites.
If we start with an atom at position ra , then we can construct the position dependent
magnetization as
X
−1
(6.3)
M (r) =
R̃p χa R̃p H ρ r − R̃p ra − tp ,
p
where ρ (r) is the distribution of the magnetic moment about the lattice site. The sum
over p is the sum over all Ng symmetry operations for the space group that describes the
lattice. Fourier transforming this equation yields [35]
M (q) =
X
1
f (q)
R̃p χa R̃−1
H
exp
iq
·
R̃
r
+
t
p a
p ,
p
Na
p
(6.4)
R∞
where f (q) = 0 exp (ikr) 4πr2 ρ (r) dr, assuming spherical symmetry for ρ (r), is the
magnetic form factor. Na is the number of operators q for which R̃q ra + tq = ra and
Ng /Na is the multiplicity of the lattice site a [35].
Next we examine the form of the equation for polarized neutron scattering intensity
[35]
I (q) = N 2 + 2P0 · (N 0 M⊥0 + N 00 M⊥00 ) + M⊥2 ,
(6.5)
where N = N 0 + iN 00 is the nuclear structure factor and P0 is the polarization vector [35].
M⊥ = M⊥0 + iM⊥00 = q × M (q) × q is the component of the reciprocal space magnetization
perpendicular to the scattering wave-vector q [35]. This equation directly relates the
reciprocal space magnetization M (q) and the polarized neutron scattering I (q), so that
it should be possible to extract one from the other.
According to Ref. [35], the actual quantity measured in experiments used to determine
χa is the polarized neutron scattering flipping ratio R = I + /I − . I + (q) and I − (q) are
the scattering intensities for neutrons polarized parallel and anti-parallel to the external
magnetic field. The ratio R can be directly related to χa using Eqns. (6.4) and (6.5). Using
numerical refinement methods it is then possible to vary the components of χa until the
flipping ratio for multiple points in reciprocal space have been matched and thus determine
the form of χa [35].
74
6.2
Calculating the Local Susceptibility
The obvious method of choice for computing the local susceptibility for Yb2 Ti2 O7 from
our model is the random phase approximation (RPA), as discussed in Chapter 3.3. If we
were to use this method, χa is given by
X
χa uv =
χuv
(6.6)
ab (q = 0, ω = 0) ,
b
where χuv
ab (q = 0, ω = 0) is the wave-vector dependent RPA susceptibility at q = 0, defined
in Eqn. (3.61). The global susceptibility is given by
χuv =
1 X uv
χa .
4 a
(6.7)
This allows for the direct calculation of the bulk susceptibility χ using the RPA methods
of Chapter 3.3.
Unfortunately the method of Chapter 3.3 and the CEF states of Appendix A do not
account for an applied magnetic field. In Ref. [19] it is explained that χa is in fact measured
in a 1 T field applied along the [110] direction. This field is large so calculations performed
in zero field will not be valid for comparison to experiment. It is possible to perform RPA
calculations of the local susceptibility using CEF states diagonalized in the presence of the
1 T field along [110], but this will yield the change in the local moment due to a small
perturbing field applied in addition to the 1 T field. The quantity measured in experiment
on the other hand is the change in the local moment due to the entire applied 1 T field.
This difference between the RPA susceptibility and the experimentally measured quantity
means that we cannot use the RPA susceptibility to compute χa . What we must actually
compute is the expectation value of the magnetic moment at each sublattice site. Dividing
this value by the applied magnetic field will yield magnetic susceptibility in units of µB /T
due to the entire applied field, and not just a weaker applied field superimposed on the 1
T field. All of the measurements of χa were collected well above the temperature of the
phase transition in Yb2 Ti2 O7 , Tc ∼ 220 mK, so the expectation value of the moments will
be zero when there is no applied field, and thus it is valid to assume that any moment
found is due to the presence of the field. To calculate the the expectation value of the
magnetic moment at each sublattice site we choose to use a mean field method.
The strong [110] field also introduces a second problem; it breaks the symmetry of
the pyrochlore lattice, and splits the doublet ground state of Yb2 Ti2 O7 . This means that
some of the assumptions made in Chapter 6.1 about the form of the local susceptibility are
75
incorrect. Specifically, the assumption of Ref. [19] that the local susceptibility expressed in
local coordinates is the same at each sublattice site, is now incorrect. This occurs because
the field splits the lattice into two types of nearest-neighbour chains, known as α and β
chains, parallel and perpendicular to the applied field respectively. These chains are shown
in Fig. 6.1. This assumption does not affect our calculations because we are not performing
a fit to experiment, but rather an independent calculation based on our model, of χa at
each sublattice site, with no assumptions about the form of this quantity,
4
3
1
2
Figure 6.1: The tetrahedral sublattice of the pyrochlore lattice, showing the local ẑ axes
~ (yellow arrow). The α
(red arrows) and the direction of the applied [110] magnetic field B
and β chains parallel and perpendicular to the applied field are labeled.
To calculate the local susceptibility using the more correct mean field method, we
compute the moments at each sublattice site in a local mean field theory. This local
mean field calculation was performed by diagonalizing the CEF matrix in the presence of
a magnetic field, in a similar manner to that used to obtain the CEF states in Appendix
A [38].
The interactions between the magnetic moments Jai are handled in the following manner. The interaction between two components of nearest-neighbour ions Jai and Jbj is
expanded in terms of expectation values as [38]
Jiau Jjbv = Jiau hJjbv i + hJiau iJjbv − 2hJiau ihJjbv i + (Jiau − hJiau i) Jjbv − hJjbv i ,
(6.8)
where hJiau i is the expectation value of the u component of the angular momentum operator
Jai . By dropping the final term of this equation, the fluctuation term, we obtain a decoupled
mean field equation. This allows us to treat the interactions between the lattice sites i, a
and j, b as an effective field. The u component of the resulting interaction field at site i, a
76
due to the v component of Jjb is then given by
˜ bv
hau
i,eff = J hJj i,
(6.9)
where J˜ is whatever combination of coupling terms in {Je } governs the interactions between these two spin components. The total interaction field at site i, a is then given by
summing over all of the nearest-neighbours j, b and components v. hai,eff is then used in
the mean field interaction portion of the Hamiltonian
Hint,MF (i, a) = −Jai · hai,eff .
(6.10)
Single ionP
wave-functions |νi, at each lattice site i, a are then determined by diagonalizing HMF = i HMF (i, a), where the sum over i, a represents the sum over all pyrochlore
lattice sites. The combination of the CEF, applied magnetic field, and interaction leads to
three terms in HMF
HMF = Hcf + HZ + Hint,MF ,
(6.11)
where Hcf is the crystal field term, discussed in Appendix A, Hint,MF is the mean field
interaction Hamiltonian, and HZ is the Zeeman interaction term, given by
X
X
HZ =
−µB gJ
Jai · B.
(6.12)
i,a
hJiau i is then given by
T r (Jiau exp (−βHMF (i, a)))
,
(6.13)
Z
where Z = T r (exp (−βHMF (i))) is the partition function. Repeated diagonalization of
HMF on a 16-site unit cell using periodic boundary conditions [38] refines the values of
hJiau i until they no longer change by a significant amount after each step. The resulting
value of hJiau i is used to determine the components of χa . The use of a 16-site unit cell
is justified by our findings in Chapter 3; that the mean-field ordering wave-vector of these
models is q = 0 [38]. This process was performed over a range of temperatures to produce
data for comparison to experiment. These calculations of χa using the mean field method
were performed by Dr. Paul McClarty, and we are very grateful for his contribution.
hJiau i =
6.3
Results
If we compute χa using only the CEF parameterizations of Refs. [19] and [20], and not
interactions, we obtain the results shown in Fig. 6.2. In Fig. 6.2 it can be seen quite clearly
77
that the values of χ⊥ fit the experimental data well despite a lack of magnetic interactions.
This CEF only model is successful in fitting χ⊥ except at the lowest temperature point
T = 2 K, where the model data falls below the experimental data point. The model values
of χk on the other hand are seen to be quite different from both experiment and each other.
The calculations of χk based on the CEF parameterization of Ref. [19] fit the experimental
data in the range T = 20 − 70 K, but the fit fails at both high and low temperatures. The
calculations based on the CEF parameterization of Ref. [20] do not fit the experimental
values of χk well at any temperature and because of this poor fit we discard this CEF
parameterization from further comparison to experiment. χk for sublattices 3 and 4 is not
shown because this quantity is zero. This is because in the β chains (see Fig. 6.1), the
applied field polarizes the magnetic moments along the field, and in the easy plane of the
ground state doublet, so there is no component of the moment along the local z axis with
which to compute χk .
Experimental χ||
Experimental χ||
Experimental χ⊥
0
10
Sublattice 1 & 2 CEF χ||
Sublattice 1 & 2 CEF χ⊥
−1
−1
10
−2
10
10
−2
10
10
(a)
Experimental χ⊥
0
10
Sublattice 1 & 2 CEF χ||
20
10
70 100
20
70 100
(b)
Figure 6.2: Calculations of the two components of χa with no interactions, using the CEF
parameterizations of Ref. [20] (a) and Ref. [19] (b). χ⊥ , and χk are shown along with the
experimentally determined values from Ref. [19]. χk for sublattices 3 and 4 is zero.
Fig. 6.3 shows calculations based on the exchange couplings {Je } of Table 4.3 for
Ref. [19], along with calculations base on the isotropic exchange model discussed in Chapter 4.2. Also shown is a reproduction of the fit of Ref. [19]. This fit fails to take into
account the sublattice structure of the pyrochlore lattice. It can be seen in Fig. 6.3 that
the incorporation of anisotropic bilinear exchange into the mean field calculations greatly
78
improves the low temperature fit to χk , leading to a good fit to all but the lowest temperature point. For comparison, the same calculations were performed using an isotropic
exchange model, constrained by the experimental value of θCW = 0.75 K [20]. The results
of these calculations are shown in Fig. 6.3, where it can be seen that these calculations do
not fit χk as well as the anisotropic exchange calculations. The fit of Ref. [19] is shown in
Fig. 6.3(c), and comparing this to the results of our anisotropic exchange calculations it
can be seen that our calculations fit the data at least as well.
So far we have only discussed the fit of our anisotropic exchange model to χa at low
temperatures. The calculations of χa when anisotropic exchange is included do not fit
the high temperature data (T > 70 K) any better than those performed with only the
CEF and Zeeman terms in HMF . One possible reason for this is that both of the CEF
parameterizations used in this work fail to correctly describe the excited states of the CEF
and so they fail at high temperatures. This question of whether the CEF parameterization
is at issue can only be answered by further experiments to determine the precise excited
states (i.e. their energies and wavefunctions) of the CEF.
RPA calculations of the powder susceptibility have also been performed using the
anisotropic exchange models defined by {Je } in Table 4.3 for the CEF parameterization
of Cao et al. [19]. The RPA calculation using the CEF states of Appendix A is valid in
this case because these experiments were performed in very low field. The results of these
calculations are shown in Fig. 6.4(b). These are actually plots of the average of the three
elements of χ (Eqn. (4.5)), because the experiments of Ref. [34] were performed on powder
samples of Yb2 Ti2 O7 . It should be noted that Ref. [34] makes no mention of any demagnetization correction in the analysis of the data. Examining these results, it can be seen that
once again, our anisotropic exchange model fits the experimental data well over a broad
range of temperatures. The comparison to experiment is made easier by Fig. 6.4(c), which
shows the differences between χ computed using: our anisotropic exchange model and experiment, our isotropic exchange model and experiment, and the fit of Ref. [34] (shown
in Fig. 6.4(a)) and experiment. From this data we can see that the isotropic exchange
calculations fit increasingly poorly at high temperatures, whereas calculations performed
using our anisotropic exchange model give χ greater than experiment below T ∼ 20 K
and less than experiment above that temperature. The fit of Ref. [34] shows the opposite
behaviour to our anisotropic exchange calculations.
79
Experimental χ
||
Experimental χ
⊥
Anisotropic Exchange Sublattice 1 & 2 χ||
Anisotropic Exchange Sublattice 1 & 2 χ
0
10
⊥
Anisotropic Exchange Sublattice 3 & 4 χ
⊥
−1
10
−2
10
10
20
70 100
Experimental χ||
Experimental χ⊥
Isotropic Exchange Sublattice 1 & 2 χ||
0
Isotropic Exchange Sublattice 1 & 2 χ
10
⊥
Isotropic Exchange Sublattice 3 & 4 χ
⊥
−1
10
−2
10
10
20
70 100
Experimental χ||
Experimental χ
⊥
Fit of Cao et al. to χ||
0
10
−1
10
−2
10
10
20
70 100
Figure 6.3: Calculations of the two components of χa with a model including magnetic interactions, using the exchange couplings {Je } from Table 4.3 for the CEF parameterization
of Ref. [19] (a). Calculations performed using Isotropic exchange and long-range dipolar
interactions Jiso = 0.06 K) are shown in panel (b). Panel (c) shows the fit of Ref. [19] to
χ⊥ for comparison purposes. χ⊥ for sublattices 3 and 4 is zero.
80
(a)
25
Powder Data
Cao et al. Fit
20
15
10
5
0
0
5
10
15
20
25
30
(b)
25
Powder Data
Isotropic Exchange
Model Hamiltonian
20
15
10
5
0
0
5
10
15
20
25
30
(c)
0.6
Cao et al. Fit
Isotropic Exchange
Model Hamiltonian
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
5
10
15
20
25
30
Figure 6.4: Panel (a) shows the experimentally measured bulk susceptibility of Yb2 Ti2 O7
from Ref. [34], along with the fit of Ref. [34] to said data. Panel (b) shows the results of our
RPA calculations using both anisotropic exchange, specifically the set of {Je } determined
using the CEF of Cao et al., and isotropic exchange with long-range dipolar interactions.
Panel (c) shows the differences between the bulk susceptibility computed using all three
models and experiment.
81
6.4
Summary
In this chapter we have presented calculations of the local and bulk susceptibility performed
using our anisotropic bilinear nearest-neighbour exchange models determined from fitting
the quasi-elastic neutron scattering. With no adjustment of parameters, the anisotropic
exchange model determined using the CEF paramterization of Ref. [19] fits experimental
measurements of χa fairly well. The CEF parameterization of Ref. [20] was found to fit
experiment poorly, perhaps indicating that future work should not be focused on this CEF
parameterization. Bulk susceptibility calculations based on our anisotropic exchange model
also fit experimental measurements of this quantity, reported in Ref. [34], quite well. These
results provide additional confirmation that our model successfully describes the magnetic
interactions in Yb2 Ti2 O7 at low temperatures.
82
Chapter 7
Ongoing and Prospective Future
Research
The next logical step after determining the magnetic interaction Hamiltonian for Yb2 Ti2 O7 ,
is to determine what the low temperature phase of this model is. In this chapter we provide
a brief description of ongoing as well as proposed work involving the low temperature
phase of this specific model for Yb2 Ti2 O7 . We also provide a brief discussion on potential
investigations of the general {Je } interaction space for the local planar but not perfectly
XY model on the pyrochlore lattice and discuss ongoing investigations of the properties of
the Heisenberg ferromagnet on the pyrochlore lattice.
7.1
Monte Carlo Simulations
Classical Monte Carlo simulations based on the anisotropic exchange model we have determined from fitting quasi-elastic neutron scattering measurements are currently ongoing.
These simulations are being performed using an effective spin-1/2 model equivalent of our
bilinear exchange model discussed in Appendix H. The classical Monte Carlo code used was
written by Dr. Pawel Stasiak for his Ph.D. Thesis [56] and takes advantage of parallel tempering to improve the equilibration of the simulation. This code has been augmented by the
inclusion of the symmetry allowed nearest-neighbour bilinear exchange terms discussed in
this study. These simulations have been performed both with and without long-range magnetic dipolar interactions. This was done because long-range magnetic dipolar interactions
significantly slow the simulation process. Figure 7.1 shows the sublattice magnetization,
83
energy per moment, and specific heat per moment for an L = 5 system with periodic
boundary conditions containing 2000 spins. L is the number of 16 site cubic units cells in
each of the cartesian coordinate directions, so that the total system is an L × L × L cube.
These results show evidence of a phase transition for both models around Tc ∼ 130 mK,
which is much closer to the experimentally observed transition temperature of 214 mK [16]
than the critical temperature found by RPA calculations in Chapter 3, TRPA ∼ 1.17 K.
This massive reduction in Tc shows that thermal fluctuations have a significant effect in this
model. Whether these transitions are first or second order is currently under investigation.
Finite size analysis and examination of the energy histogram of the simulations have yet
to provide concrete evidence for a first or second order phase transition for either model.
It is interesting to note the significant difference in the height and shape of the peak in the
specific heat computed from the two models in Fig. 7.1(c). This indicates that the phase
transition is somehow different without the presence of long-range magnetic dipolar interactions than with them. The peak in the specific heat for the model with no long-range
magnetic dipolar interactions is rather small and rounded. This is very strange because
studies of the low temperature phase of this model discussed in the next section indicate
that this phase transition is likely an example of a thermal order-by-disorder transition.
Theoretical work to date has found that such phase transitions should be at least weakly
first order [57], which is inconsistent with the observed peak in the specific heat, though
simulations of larger system sizes my resolve this issue.
−0.25
0.45
D=0 K
D=0.01848 K
0.4
6
D=0 K
D=0.01848 K
0.35
D=0 K
D=0.01848 K
5
−0.3
4
0.3
0.2
−0.35
Cv (J /K)
M
E (K)
0.25
−0.4
3
2
0.15
0.1
−0.45
1
0.05
(a)
0
0.1
0.15
0.2
T (K)
0.25
−0.5
0.1
(b)
0.15
0.2
T (K)
0.25
(c)
0
0.1
0.15
0.2
0.25
T (K)
Figure 7.1: Monte Carlo simulation results for an L = 5 system with periodic boundary
conditions. Panel (a) shows the sublattice magnetization, which goes to 1/2 as T → 0, as
expected for a classical spin-1/2 system. Panel (b) shows the energy per magnetic moment
and panel (c) shows the specific heat per magnetic moment. All three of these quantities
show features near Tc ∼ 130 mK, indicating the presence of a phase transition.
Figure 7.2 shows the ground states of the simulations performed with and without long84
range magnetic dipolar interactions. These states were determined with the assistance of
mean field calculations, discussed in Chapter 7.2. The ground state of our anisotropic
exchange model without long-range magnetic dipolar interactions has no net moment. The
ground state when long-range magnetic dipolar interactions are used with D = 0.01848 K,
is a ferromagnetic, canted spin ice state state, with a net moment. This begs the question:
why is the ground state of our anisotropic exchange model for Yb2 Ti2 O7 so sensitive to
such a weak interaction? D is approximately 1/10th of the strength of the next strongest
interaction in this model, and this relative weakness would naively lead one to expect it
to have no effect on the ground state of the system, making its large effect on the ground
state of the model very curious.
(a)
(b)
Figure 7.2: The low temperature ground states of the classical anisotropic spin-1/2 model
determined using Monte Carlo simulations and mean field calculations. Panel (a) shows
the ground state of the model with no long-range magnetic dipolar interactions. Panel (b)
shows the ground state when D = 0.01848 K.
7.2
Mean Field Calculations
In addition to classical Monte Carlo simulations, mean-field calculations of the free energy
have been performed using the effective spin-1/2 equivalent to our model by Mr. Behnam
Javanparast. These calculations confirm the results of the Monte Carlo simulations, though
they do find that the ground state of the model without long-range magnetic dipolar
85
interactions is degenerate to symmetry rotation within the easy planes. The Monte Carlo
simulations found that specific orientations of the moments were selected, such as the one
shown in Fig. 7.2(a), suggesting that thermal order-by-disorder 3 is taking place. This has
been confirmed by histogram analysis of the easy plane moment angles from Monte Carlo
simulations shown in Fig. 7.3. There it can be seen that the in plane moments prefer to
lie at angles φ = (n + 1/2)(π/3). These angles correspond to the ψ3 state of Ref. [58], also
known as the “anti-Palmer-Chalker” state, which is shown in Fig. 7.2.
0.02
T = 0.01 K
T = 0.02 K
T = 0.03 K
0.015
0.01
0.005
−120
−60
0
60
120
Figure 7.3: Monte Carlo calculations of P (φ) for our bilinear exchange model with no
long-range magnetic dipolar interactions for T = 0.01 K, 0.02 K, and 0.03 K for an L = 2
system. φ is the easy plane rotation angle, the out of plane angle θ is 0.
Mean field calculations have also been performed to attempt to understand the sensitivity of our anisotropic exchange model to the presence of the long-range magnetic dipolar
interaction. The result of these calculations is that the eigenvalues of the Fourier transformed interaction matrix (see footnote on pg. 62) corresponding to the normal mode
amplitudes associated with the canted spin ice ground state and the anti-Palmer-Chalker
3
Thermal order by disorder is a phenomenon whereby unique ground states are selected by thermal
fluctuations from a manifold of extensively degenerate classical ground states [59]. The specific states
selected by this mechanism will be those with the greatest density of zero modes. For these states, the
magnitude of the entropic contribution to the free energy will be greater than for the other states in the
manifold, leading to the minimization of the free energy in these states.
86
ground state are very close close together in energy (∆ ∼ 3%) when there are no longrange magnetic dipolar interactions. As the strength of the long-range magnetic dipolar
interactions is increased, the eigenvalues corresponding to the two different ground states
cross, leading to the change in the ground state. This crossing occurs at Dc ∼ 0.008 K,
which is approximately half of D = 0.01848 K the value of D for Yb2 Ti2 O7 . The small gap
between the eigenvalues associated with two very different types of magnetic LRO when
no long-range magnetic interactions are present means that this specific model will be very
susceptible to weak perturbations. This explains why such weak long-range magnetic interactions are capable of driving a phase transition from the anti-Palmer-Chalker state to
the canted spin ice state. The fact that the correct value of D for Yb2 Ti2 O7 is approximately twice the critical value Dc , means that Yb2 Ti2 O7 is well within the canted spin
ice phase in the mean field phase diagram of the nearest-neighbour bilinear exchange with
long-range magnetic dipolar interaction, {Je } + D, model of the XY like effective spin-1/2
model of Yb2 Ti2 O7 . This is a very qualified statement, because many different properties
of the model, such as the planar nature of the spins and the specific strength of all of
the interactions, combine to yield this phase. It does however provide some motivation
for further explorations of this phase diagram, to determine if any other combinations of
anisotropic exchange, long-range magnetic dipolar interactions, and anisotropic spins yield
interesting physics.
7.3
Quantum Fluctuations
We saw from the classical Monte Carlo results that thermal fluctuations are very important
in our anisotropic exchange model, leading to a massive change in the critical temperature
of the model computed using RPA and Monte Carlo methods. At very low temperatures
these thermal fluctuations will be greatly reduced but they may be replaced by quantum
fluctuations of the magnetic moments. Yb2 Ti2 O7 may be especially susceptible to such
quantum fluctuations as the ground state doublet can be treated as an effective spin-1/2
model and spin-1/2 systems have been known to exhibit large quantum fluctuations. The
low temperature phase of the model with no long-range magnetic dipolar interactions is
anti-ferromagnetic in nature suggesting that quantum fluctuations may be significant in
this phase. The low temperature phase of the model with long-range magnetic interactions
on the other hand, is ferromagnetic in nature and has a discrete set of ground states,
suggesting that quantum fluctuations should be gapped out. The size of this gap, and thus
its ability to suppress quantum fluctuations is an outstanding question of this model. The
strength of the long-range magnetic dipolar interaction that drives the transition from the
87
anti-Palmer-Chalker phase, found when no long-range magnetic dipoles are present, to the
canted spin ice phase is very small, D ∼ 0.02 K. This means that the energy gap associated
with the lowest energy spin wave mode may be quite small, so that quantum fluctuations
may in fact be significant in the canted spin ice phase. Experiments performed on the
low temperature phase of Yb2 Ti2 O7 find no evidence of magnetic long-range order [6, 15]
(see footnote on pg. 15). Perhaps quantum fluctuations drive the system into a disordered
phase or even a spin liquid phase, thus explaining the lack of magnetic long-range order
in Yb2 Ti2 O7 at low temperatures. Investigations using the method of Ref. [60] for spin
wave calculations on the pyrochlore lattice have begun in order to determine what effects
quantum fluctuations play in the ground state of our model, but so far no results have
been obtained.
7.4
Ground States the Full {Je} Phase Space for the
Non-Ideal Local-Planar Pyrochlore Model
The small difference between the mean field eigenvalues of the Fourier transformed interaction matrix (see footnote on pg. 62) when long-range magnetic dipolar interactions are
not present, which correspond to two very different types of ground state of our anisotropic
exchange model, is very intriguing. It raises questions about the stability of the ground
state to changes in the relative strengths of the bilinear exchange terms {Je }. Investigation
of the more general “non-perfect XY” model in the space of all possible {Je } may yield
interesting results. Particularly, as the long-range magnetic dipolar interaction has been
found to be very significant in determining the low temperature phase of the model, perhaps investigations of the strength of the pseudo-dipolar exchange interaction as a function
of the local Ising exchange interaction strength may yield interesting results. It may be
possible to drive the anti-Palmer-Chalker to canted spin ice phase transition using some
combination of the bilinear exchange interactions, and as the pseudo-dipolar exchange is
so similar in form to the long-range magnetic dipolar interaction, this seems an obvious
place to begin the investigation.
Also, as previously mentioned, experiments on Yb2 Ti2 O7 find no evidence of longrange magnetic order in the low temperature phase, suggesting some type of disorder is
preventing the system from achieving this ordered phase. One possible source of this
disorder is material imperfections not considered in this work. Theoretical studies of the
sensitivity of our model or similar models to dilution of the magnetic moments, and or
distortion of the lattice may also yield very interesting results.
88
7.5
Rods of Scattering: The Pyrochlore Heisenberg
Ferromagnet
Finally, preliminary investigations have been performed on the Heisenberg ferromagnet on
the pyrochlore lattice. These investigations were originally performed as a simple test case,
in order to better understand the real space correlations that arise from our anisotropic
exchange model, discussed in Chapter 5. This investigation is based on computing the
quasi-elastic neutron scattering and real space correlation functions using the mean-field
method presented in Refs. [54, 55, 61]. This method is based on finding the normal modes
of the Fourier transformed order parameters for each of the sublattice sites (see footnote
on pg. 62). From the eigenvalues and eigenvectors of the Fourier transformed interaction
matrix K(Q) (Eqn. (3.61)) corresponding to these normal modes, the mean-field neutron
scattering and real-space correlation functions can be computed. This mean field approach
is used as we do not possess a set of crystal field states for the Heisenberg ferromagnet,
which are a required starting point for the use of the RPA method discussed in Chapter
3.3.
The results of our mean field calculations of the quasi-elastic neutron scattering and
real space correlation functions, S(r) (Eqn. (5.2)), can be seen in Fig. 7.4. Figures 7.4(a)
and (b) show that the Heisenberg ferromagnet on the pyrochlore lattice displays [111] rods
of quasi-elastic neutron scattering intensity. Figure 7.4(c) shows the real space correlation
function S(r) computed along three of the high symmetry directions of the pyrochlore
lattice. From this calculation we can see that S(r) for the Heisenberg ferromagnet shows
similar splitting between in the strength of S(r) along the nearest-neighbour direction [01̄1]
and the second nearest-neighbour [12̄1] to the splitting found for our anisotropic exchange
model (see Fig. 5.3). This indicates that the presence of stronger correlations along nearestneighbour chains within the kagome planes that make up the pyrochlore lattice than along
other directions within these planes may be responsible for rod-like features in the neutron
scattering of pyrochlore materials. In the case of the Heisenberg ferromagnet the splitting
is not as large as that observed in our anisotropic exchange model. This weaker splitting
may explain why the rods are weaker in Figs. 7.4(a) and (b) than those found for our
anisotropic exchange model. There is one significant difference between the correlation
functions of the two models, the fact that for the Heisenberg ferromagnet, S(r) computed
along [111] is just as strong as S(r) computed along the nearest neighbour chains direction.
What this means in relation to form of the rod-like features observed is currently unknown.
The rods of scattering generated by the Heisenberg ferromagnet on the pyrochlore
lattice, and the splitting of S(r) between nearest-neighbour chains and second nearest89
neighbour directions, suggest that there may be a hidden relationship between these two
models. Perhaps studying the Heisenberg ferromagnet may lead to a better understanding
of how the rods of scattering present in Yb2 Ti2 O7 arise, and how the rods of scattering are
related to the magnetic correlations in Yb2 Ti2 O7 . The existence of [111] rods in the neutron
scattering of the Heisenberg ferromagnet on the pyrochlore lattice may also indicated that
the rods are directly related to the symmetry of the pyrochlore lattice. Further study
of the Heisenberg ferromagnet on the pyrochlore lattice is required and a real material
described by this model would be quite interesting to study. One current prospect for such
a material is Lu2 V2 O7 , but this material has recently been found to possess significant easy
axis anisotropy [62].
90
Figure 7.4: Mean field neutron scattering results for the Heisenberg ferromagnet at (a)
T = 1.1 TcMF and (b) T = 1.05 TcMF . Panel (c) shows real space correlation functions
computed at T = 1.05 TcMF plotted along the high symmetry directions of the pyrochlore
lattice, as shown in the inset.
91
Chapter 8
Conclusions
The study of frustrated magnetic materials has led to the discovery of many novel and exotic
low temperature phenomena. One family of frustrated magnetic materials in particular,
the rare earth pyrochlore oxides, have been found to display a broad range of interesting
low temperature behaviours ranging from spin liquid behaviour in Tb2 Ti2 O7 [9], to spin ice
in Dy2 Ti2 O7 and Ho2 Ti2 O7 [1], to persistent spin dynamics at even the lowest measured
temperatures in Gd2 Sn2 O7 and Er2 Ti2 O7 [1]. These interesting phenomena arise due to
material specific conditions, making the study of other rare earth pyrochlore oxide materials
an excellent prospect for discovering even more exotic low temperature phenomena. One
such material is Yb2 Ti2 O7 , where experiments have found evidence of a first order phase
transition [15, 16] in powder samples of this material at Tc ∼ 220 mK, but no evidence
of long-range order below the temperature of this phase transition [6, 15] (see footnote on
pg. 15). In fact evidence is found for continued spin dynamics below the temperature of the
phase transition [15]. In order to understand what is actually occurring at low temperatures
in this material, a model that describes all of the contributions to the environment inhabited
by the magnetic moments, including the crystal field, exchange interactions, and longrange magnetic dipolar interactions is required. The crystal field of Yb2 Ti2 O7 has been
determined experimentally in Refs. [19, 20], and the strength of the magnetic dipolar
interactions are fixed by the fact that the Yb3+ ion is a J = 7/2 ion, leaving only the
exchange interaction contribution to this model as an unknown. In this work we have set
out, and as we have seen, succeeded, in determining a set of exchange interactions that
successfully describe many experimentally observed properties of Yb2 Ti2 O7 .
The model we have chosen to describe the exchange interactions between the magnetic
moments of the Yb3+ ions in Yb2 Ti2 O7 includes all nearest-neighbour symmetry allowed
bilinear exchange model which, on the pyrochlore lattice, consists of four terms. These
92
four terms can be expressed in many different ways, but the easiest form to understand is
one that consists of local Ising exchange, isotropic exchange, pseudo-dipolar exchange, and
Dzyaloshinskii-Moriya exchange. This model choice leaves us with four free parameters,
which are the strengths of the four different types of exchange.
In order to determine the strengths of these four type of exchange interactions in Yb2 Ti2 O7 , we have performed a simulated annealing fit to quasi-elastic neutron scattering,
which was collected well above the temperature of the experimentally observed phase transition. This neutron scattering data shows rods of scattering intensity along the [111]
crystallographic direction, indicating the presence of interaction driven correlations between the Yb3+ magnetic moments, and making it an excellent quantity from which to
determine the form of the magnetic interactions, as it is so highly featured. The fit to
this data was performed by comparing random phase approximation (RPA) neutron scattering calculations performed using our proposed model to experimental observations of
the neutron scattering intensity along cuts through the (h, k, k) plane. This difference was
treated as an effective energy in a simulated annealing optimization of the strengths of
the four interactions, along with a contribution to ensure agreement between the experimental and model Curie-Weiss temperatures. These calculations successfully determined
two sets of exchange strengths, one for each crystal field parameterization used, that reproduce the experimental neutron scattering to a high degree of accuracy. Both of these
models have ferromagnetic local Ising exchange as the energetically strongest interaction,
which is interesting as this is opposite to the local planar nature of the crystal field ground
state of Yb2 Ti2 O7 . The other three exchange terms are found to be approximately four
times weaker than the local Ising exchange, but still energetically significant. The fact
that the Dzyaloshinskii-Moriya interaction is so strong is interesting, as this interaction
arises as a perturbative effect due to spin-orbit coupling in magnetic materials, making
it typically very small. Recent work published in Ref. [52] has shown that such a large
Dzyaloshinskii-Moriya interaction is in fact allowed in Yb2 Ti2 O7 . The set of interactions
we have determined does suffer from some limitations. The model considered only takes
into account bilinear exchange terms, while Yb3+ ions in Yb2 Ti2 O7 are in fact J = 7/2 ions,
which means that higher order exchange terms are allowed in this system. Our model is an
“unprojected” full blown angular momentum, J, form of an effective spin-1/2 model of the
interactions in Yb2 Ti2 O7 . Effective spin-1/2 models involve “down projecting”, projecting
all of the interactions in the microscopic Hamiltonian, Hmic , into a Hilbert space spanned
by the ground state doublet of the crystal field. Such a down projection is allowed because
of the fact that the excited states of the crystal field are at significantly higher energies
than those of the interactions, and also much larger than the temperature at which the
neutron scattering was collected. This means our model is most likely only correct at low
93
energies, and not at energies on the scale of the crystal field splitting.
After successfully finding a set of exchange interactions capable of the describing the
quasi-elastic neutron scattering of Yb2 Ti2 O7 , we then proceeded to use this model to
compute the real space correlations generated by this model in an effort to understand
how the rods of scattering intensity arise from our interaction model. The real space
correlation function was computed by integrating the RPA susceptibility over the first
Brillouin zone of the FCC lattice. The real space correlations were computed along three
high symmetry directions on the pyrochlore lattice: the nearest-neighbour chain direction,
the second nearest-neighbour direction, and the cubic unit cell body diagonal direction.
We found that the correlations along the second nearest-neighbour and cubic unit cell
body diagonal directions are weaker than the correlations along nearest-neighbour chains.
This indicates that the rods are due to the presence of strong correlations along all of the
nearest-neighbour chains in the pyrochlore lattice, not isotropic correlations within kagome
planes in the pyrochlore lattice, as has been suggested in Ref. [40].
Finally we computed the bulk and local susceptibilities from our model using mean
field and RPA methods. These calculations were compared to experimental measurements
of these quantities reported in Refs. [19, 34]. We found good agreement between our model
and experiment, but only using the crystal field parameterization of Ref. [19]. The crystal
field parameterization of Ref. [20] resulted in a very poor fit to the experimental local
susceptibility, so it was eliminated from any further consideration. The success of the
anisotropic exchange model corresponding to the crystal field parameterization of Ref. [19]
provides additional confirmation that our bilinear exchange model does indeed describe the
magnetic interactions of Yb2 Ti2 O7 , at least at low energies. The model fails to fit the local
susceptibility at high temperatures suggesting that the crystal field parameterization may
incorrectly describe the excited crystal field states. The bulk susceptibility computed using
our anisotropic exchange model was also found to fit experiment well at low temperatures.
With a Hamiltonian describing the magnetic interactions in Yb2 Ti2 O7 now available,
it should be possible to use it to better understand the low temperature phase of this material. We have presented some preliminary results aimed at determining the ground state
of our model, using both classical Monte Carlo simulations and mean field calculations.
These results show that the ground state is highly dependent on the presence of long-range
magnetic dipolar interactions, suggesting that our model is highly susceptible to perturbations. This may indicate that quantum fluctuations or material disorder could destroy
the ordered ground state of the classical spin-1/2 model, and lead to the experimentally
observed fluctuating and disordered low temperature phase. This possibility means that
calculations of the quantum fluctuations of the computed ground state may yield very
interesting results. Further calculations to determine the nature of the phase transition of
94
this model are required, as calculations to date have failed to determine concretely whether
this phase transition is first or second order. The possibility of an order-by-disorder phase
transition in the case where long-range magnetic dipolar interactions are not present has
been found, and further investigations of this transition are also required. The sensitivity
of the ground state of our model for Yb2 Ti2 O7 to small perturbations should also motivate
investigations into the ground states of the more general local planar pyrochlore system.
Various combinations of all of the possible symmetry allowed nearest-neighbour exchange
interactions may yield very exotic phenomena.
95
APPENDICES
96
Appendix A
Crystal Field Parameterizations
This Appendix contains a detailed description of the CEF parameterizations of Yb2 Ti2 O7 used in this study. Two such parameterizations are used, that of Hodges et al. from
Ref. [20], described in Section A.1 and that of Cao et al. from Ref. [19], described in
Section A.2. These crystal field parameterizations take the from
(A.1)
where Onm are Steven’s operators, which are defined in Appendix B.
A.1
The Crystal Field of Hodges et al.
In this section we provide the details of the crystal field parameterization of Ref. [20]. This
crystal field parameterization is based on 170 Yb Mössbauer spectroscopy, 170 Yb perturbed
angular correlations, specific heat measurements, and magnetic susceptibility measurements [20]. The crystal field parameters determined based on these measurements are
shown in Table A.1. Using these parameters HCF can be diagonalized [63] to determine
the energies and wave-functions of the crystal field states; these results are shown in Table
A.2. It should be noted that the ground state doublet here is different to that published
in the literature. Reference [61] indicates that the first term of the ground state wavefunction in Ref. [20] should have a ± sign in front of it, and the small difference in the
prefactor of the |J = 7/2, Jza = +5/2i and |J = 7/2, Jza = −5/2i terms is due to small
differences in diagonalization results. As we can see in Table A.2, the separation between
the ground state doublet and the first excited state doublet is very large, so that the low
97
temperature physics of Yb2 Ti2 O7 will be governed by the ground state of this set of crystal
field parameters. Because of this it is useful to compute the ground state matrix elements
hψ01 |Jxa |ψ02 i, hψ01 |Jya |ψ02 i, and hψ01 |Jza |ψ01 i. These matrix elements are used in the creation
of the g tensor, used when projecting the system on to an effective spin-1/2 ground state,
a useful technique when the behaviour of the system is governed by the doublet ground
state. Performing this calculation, we find hψ01 |Jxa |ψ02 i = ihψ01 |Jya |ψ02 i = J⊥ = 1.83 and
hψ01 |Jza |ψ01 i = Jk = 0.772, which can be used to determine the ground state g tensor of the
Yb3+ ion via the equation [52]:


2gJ J⊥
0
0


(A.2)
g= 0
2gJ J⊥
0 
0
0
2gJ Jk
where gJ is the Landé factor for the Yb3+ ion, gJ = 8/7 [18]. This equation yields the
following g tensor for the ground state of the Yb3+ ion in Yb2 Ti2 O7 given in Table A.2:


4.18 0
0


g =  0 4.18 0 
(A.3)
0
0 1.76
Table A.1: Crystal Field Parameters for Yb2 Ti2 O7 from References [19] and [20].
Hodges et al. [20]
Cao et al. [19]
A.2
B20
12.4K
11.9K
B40
−0.178K
−0.509K
B43
−6.8K
6.147K
B60
0.021K
0.006K
B63
−0.14K
0.189K
B66
0.161K
0.098K
The Crystal Field of Cao et al.
In this section we provide details of the crystal field parameterization of Ref. [19]. This
crystal field parameterization is based on rescaling the crystal field parameters of Ho2 Ti2 O7 ,
and fitting to the 170 Yb perturbed angular correlations of Ref. [20] as explained in Ref. [19].
The crystal field parameters determined using this method are shown in Table A.1. It
should be noted that the crystal field in Ref. [19] is presented in a different notation from
98
99
|J
|J
|J
|J
|J
|J
|J
|J
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= −7/2i
= −5/2i
= −3/2i
= −1/2i
= +1/2i
= +3/2i
= +5/2i
= +7/2i
Energy (K)
0.0
−0.245
0.0
0.0
0.889
0.0
0.0
0.388
0.0
|ψi10
−0.388
0.0
0.0
0.889
0.0
0.0
0.245
0.0
0.0
|ψi20
−0.010
0.912
0.0
−0.015
0.346
0.0
0.040
−0.216
621.08
|ψi11
−0.216
−0.040
0.0
−0.346
−0.015
0.0
0.912
0.010
621.08
|ψi21
0.0
0.0
−1.0
0.0
0.0
0.0
0.0
0.0
732.97
|ψi12
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
732.97
|ψi22
0.896
0.007
0.0
0.301
−0.006
0.0
0.326
0.018
943.50
|ψi13
0.018
−0.326
0.0
0.006
0.301
0.0
0.007
−0.896
943.50
|ψi23
Table A.2: The wave-functions and energies of the crystal field states of Yb2 Ti2 O7 determined using the
th
crystal field parameterization of Ref. [20]. |ψim
degenerate wavefunction of the nth energy state,
n is the m
which is given by summing all of the terms in the associated column. The quantization direction z is chosen
to be along the local h111i direction at each sublattice site a.
Ref. [20], that of irreducible tensor operators. The conversion between the two forms is
n
performed by noting that in the notation of Ref. [19] and Ref. [20], Bnm = an Am
n hr i,
n
while in the unfortunately similar notation of Ref. [19], Bnm = dn Am
n hr i. The values of
an can be found in Table 18 of Ref. [64], and the values of dn can be found in Table 6-1
of Ref. [24]. As in the previous section, using these parameters, HCF can be diagonalized
to determine the energies and wavefunctions of the crystal field states, which are shown
in Table A.3. Similar to the crystal field described by the parameterization of Ref. [20],
the crystal field described by the parameterization of Ref. [19] has a ground state doublet
well separated from the first excited state. Due to this, the low temperature physics
of a system described by these crystal field parameters is governed by the ground state
doublet. As before we compute the ground state matrix elements hψ01 |Jxa |ψ02 i, hψ01 |Jya |ψ02 i,
and hψ01 |Jza |ψ01 i and the g tensor via Eqn. (A.2). Using the crystal field states of Table A.3,
we find hψ01 |Jxa |ψ02 i = ihψ01 |Jya |ψ02 i = J⊥ = 1.74, hψ01 |Jza |ψ01 i = Jk = 0.982, and

3.98 0
0


g =  0 3.98 0  .
0
0 2.24

(A.4)
With all of the wave-functions for the crystal field of the Yb3+ ion for both crystal field
parameterizations, our description of HCF is complete.
100
101
|J
|J
|J
|J
|J
|J
|J
|J
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= 7/2, Jza
= −7/2i
= −5/2i
= −3/2i
= −1/2i
= +1/2i
= +3/2i
= +5/2i
= +7/2i
Energy (K)
0.0
0.089
0.0
0.0
0.907
0.0
0.0
−0.411
0.0
|ψi10
0.411
0.0
0.0
0.907
0.0
0.0
−0.089
0.0
0.0
|ψi20
0.0
0.0
−0.438
0.0
0.0
0.899
0.0
0.0
693.07
|ψi11
0.0
0.0
−0.899
0.0
0.0
−0.438
0.0
0.0
693.07
|ψi21
−0.873
0.0
0.0
0.420
0.0
0.0
0.249
0.0
796.82
|ψi12
0.0
0.249
0.0
0.0
−0.420
0.0
0.0
−0.873
796.82
|ψi22
0.0
0.964
0.0
0.0
0.025
0.0
0.0
0.264
1051.02
|ψi13
0.263
0.0
0.0
−0.025
0.0
0.0
0.964
0.0
1051.02
|ψi23
Table A.3: The wave-functions and energies of the crystal field states of Yb2 Ti2 O7 determined using the
th
crystal field parameterization of Ref. [19]. |ψim
degenerate wavefunction of the nth energy state,
n is the m
which is given by summing all of the terms in the associated column. The quantization direction z is chosen
to be along the local h111i direction at each sublattice site a.
Appendix B
The Stevens-operator Equivalents
Required to Describe the Crystal
Field of Yb2Ti2O7
This appendix provides the operator equivalent definitions of the Steven’s operators [64],
used to define the crystal field of Yb2 Ti2 O7 . The crystal field of Yb2 Ti2 O7 is defined as:
(B.1)
n
n
are Steven’s operators, defined as [64]:
are coefficients, given in Table A.1. Om
where Bm
O20 = 3 (Jza )2 − J a (J a + 1)
O40 =
35 (Jza )4 −
a
a
(B.2)
30J a (J a + 1) (Jza )2
a 2
a
+
25 (Jza )2
2
− 6J (J + 1) + 3 (J ) (J + 1)
1 a a
O43 =
Jz J+ + J−a S
2
O60 = 231 (Jza )6 − 315J a (J a + 1) (Jza )4 + 735 (Jza )4
(B.3)
(B.4)
+ 105 (J a )2 (J a + 1)2 (Jza )2 − 525J a (J a + 1) (Jza )2
+ 294 (Jza )2 − 5 (J a )3 (J a + 1)3 + 40 (J a )2 (J a + 1)2
− 60J a (J a + 1)
a 3
o
1n
3
a 3
a
a
a
a
a 3
O6 =
11 (Jz ) − J (J + 1) Jz − 59Jz
J+ + J−
2
S
1
6
a 6
a 6
,
O6 =
J+ + J−
2
102
(B.5)
(B.6)
(B.7)
where {AB}S = 12 (AB + BA) [64]. The a superscript denotes the use of the local h111i
direction for the a sublattice site as the quantization direction for the total angular momentum J.
103
Appendix C
Symmetry Allowed Exchange
Interactions on the Pyrochlore
Lattice
To describe the exchange interactions present in Yb2 Ti2 O7 we propose that Hex contains all
the symmetry allowed nearest-neighbour bilinear exchange interactions on the pyrochlore
lattice. These are exchange interactions whose form remains unchanged under the symmetry operations of the pyrochlore lattice, which is described by the F d3̄m (O7h ) space group
[7] and the Oh point group. The space group describes all of the operations that leave the
lattice structure unchanged including translations, while the point group considers only
rotation, inversion, and reflection operations.
C.1
Deriving the Symmetry Allowed Exchange Interactions
In order to determine the number of independent symmetry allowed interactions that make
up J (Eqn. (2.5)) and determine the form of these interactions, we must first understand
some basic representation theory. The Oh point group is made up of symmetry operations
such as rotation and reflection, but not translation, that transform the lattice structure
into itself. These operations are broken up into classes, groups of operations that when
written in matrix form can be transformed into one another by a third symmetry operation
in the group [65]. The symmetry operations A and B are members of the same class if
104
there exists a third symmetry operation C such that C −1 AC = B. A representation
Γ is the set of matrices {D(R)} that perform all of the symmetry operations {R} of a
group [65]. A very useful property of the matrices {D(r)} is that their trace, or character
(χ(R)) as it is called in representation theory, does not change depending on the coordinate
system chosen for the representation, and every matrix D(R) in a class has the same
character. One final useful aspect of representation theory is the idea of an irreducible
representation. A representation is said to be reducible if there exists a similarity transform
that transforms all of the matrices of the representation into the same block diagonal form
[65]. A representation where this is not possible is said to be irreducible. The characters
of all of the symmetry operations of a group for all of irreducible representations are given
in character tables, such the the Oh character table shown in Table C.1.
The first step in determining the number of independent symmetry allowed nearestneighbour bilinear exchange interactions on the pyrochlore lattice is to define the order of
the group (Eqn. 3-10 in Ref.[65]),
X
lα .
(C.1)
h=
α
th
lα is the dimensionality of the α irreducible representation, which is identical to the
character of the identity symmetry operation E for the αth irreducible representation [65].
We must also define mα , the number of times the αth irreducible representation appears in
the decomposition of a reducible representation. This quantity is given by (Eqn. 3-12 in
Ref. [65])
1X α
χ (R)χ∗ (R),
(C.2)
mα =
h α
where χα (R) is the character of the symmetry operation R in the αth irreducible representation, and χ∗ (R) is the character of the symmetry operation R in the representation we
seek to decompose [65].
Now we may define our chosen representation, Γ, in which to compute the form of the
symmetry allowed interactions. We chose the representation consisting of all of the possible
bilinear interactions between the various cartesian components of the J angular momenta
at the four sublattice sites:
X
Γ=
Jua Jvb ,
(C.3)
a6=b,u,v
where u and v are the cartesian components x, y, and z. Γ encompasses all of the possible nearest neighbour interactions that may be contained in Hex . In this representation,
assuming that the interaction matrix is symmetric, that is Jua Jvb = Jvb Jua , there are 54 independent components in the representation, thus Γ has a dimensionality of 54. Now we
105
require χ(R) for all of the symmetry operations of the group Oh in this representation.
The easiest character to compute is that of the identity operation E, which trivially has a
character of 54. The characters of the other operations are more difficult to compute. In
the order of Table C.1, the symmetry operations of the Oh point group are [65]:
• E, The identity transformation,
• C3 , 2π/3 rotation about the h111i cubic body diagonals,
• C02 , π rotation about the x̂ + ŷ, x̂ + ẑ, and ŷ + ẑ directions,
• C4 , π/2 rotation about the cartesian x̂, ŷ, and ẑ directions,
• C2 , π rotation operation about the x̂, ŷ, and ẑ directions,
• i, The inversion operator that swaps x to −x, y to −y, and z to −z,
• S4 , π/2 rotation about the x̂, ŷ, and ẑ directions accompanied by a reflection in
through the plane perpendicular to these directions,
• S6 , π/3 rotation about the h111i cubic body diagonals accompanied by a ref reflection
in through the plane perpendicular to these directions,
• σh , reflection through the planes normal to the C4 rotation directions,
• σd , reflection though the planes normal to the C’2 rotation directions.
Understanding exactly how these operations act on Γ is not a trivial process, but there
are some symmetry relations that make it easier to understand how the elements of Γ
will transform into one another under these symmetry operations. These relations will
also simplify the computation of the character of each of the symmetry operations in this
representation. These symmetry relations revolve around the idea that, in Table C.1,
all of the symmetry operations in the columns to the right of the inversion operation
are operations to the left of the inversion operation, combined with inversion. In fact the
inversion operator itself can be thought of as the identity operator combined with inversion
[65]. It should be noted that the symmetry operations that result from the combination
of inversion with the elements E, C3 , C02 , C4 , and C2 are not in the same order as these
operations. The symmetry relationships are i=i⊗E, S4 =i⊗C4 , S6 =i⊗C3 , σh =i⊗C2 ,
and σd =i⊗C02 . This means that once we know how one symmetry operation acts on
Γ, we simply invert it and automatically generate another symmetry operation. Another
106
useful relation is that the character of symmetry operations related by inversion are either
identical, or multiplied by −1 depending on whether a representation is odd or even under
inversion. In the case of Γ, as inversion maps all four sub-lattices to themselves and because
all the terms are bilinear, the inversion of the coordinate system is cancelled out.
Table C.1: The character table of the Oh point group [65]. The rows give the characters of
the various irreducible representations, while the columns label the symmetry operations.
A1g
A2g
Eg
T1g
T2g
A1u
A2u
Eu
T1u
T2u
E
1
1
2
3
3
1
1
2
3
3
8C3
1
1
-1
0
0
1
1
-1
0
0
6C02
1
-1
0
-1
1
1
-1
0
-1
1
6C4
1
-1
0
1
-1
1
-1
0
1
-1
3C2
1
1
2
-1
-1
1
1
2
-1
-1
i
1
1
2
3
3
-1
-1
-2
-3
-3
6S4
1
-1
0
1
-1
-1
1
0
-1
1
8S6
1
1
-1
0
0
-1
-1
1
0
0
3σh
1
1
2
-1
-1
-1
-1
-2
1
1
6σd
1
-1
0
-1
1
-1
1
0
1
-1
Table C.2: The characters of the symmetry operations of the Oh point group in the representation Γ.
Γ
E
54
8C3
0
6C02
2
6C4
0
3C2
10
i
54
6S4
0
8S6
0
3σh
10
6σd
2
The precise action of each of the symmetry operations of the Oh symmetry group on
the elements of Γ are given in Appendix D. If we define a vector space with dimension
54, one independent dimension for each independent element in Γ, we can write all of the
symmetry operations in Γ as 54×54 matrices. The elements of these matrices describe how
the components of Γ transform into each other under the various symmetry operations of
the Oh point group. For example, under the identity operation, E, all of the elements of Γ
107
transform into themselves, so the matrix representation of E is simply a 54 × 54 identity
matrix. The character of this or any other symmetry operation is then determined by
taking the trace of the matrix representation of the symmetry operation. The characters
of all of the symmetry operations in the representation Γ, computed using this method,
are given in Table C.2.
Now that we know the character of all of the symmetry operations in Oh for the reducible
representation Γ, we can use Eq. (C.2) to decompose Γ into a combination of irreducible
representations. Using the characters of the symmetry operations of the point group Oh
for all of the irreducible representations of Oh given in Table C.1, and the characters of
the symmetry operations in the representation Γ given in Table C.2, we can compute the
number of times each of the irreducible representations appears in Γ. Performing this
decomposition we obtaining the result
Γ = 4A1g ⊕ 3A2g ⊕ 7Eg ⊕ 5T1g ⊕ 6T2g .
(C.4)
The irreducible representation A1g is the irreducible representation where all of the symmetry operations leave the elements unchanged, as the character of all of the symmetry
operations in the representation are identical. The fact that A1g appears four times in Γ
means that Γ contains four sets of elements that will be left unchanged under all of the
symmetry operations of Oh . These sets are referred to as symmetry invariant.
In order to determine what these four sets of elements are, we must turn to the
concept of symmetry coordinates in representation theory. Symmetry coordinates are
n-dimensional hyper-vectors containing combinations of the elements of a chosen representation that when acted on by any symmetry operation of a group, transform in simple ways
[65]. Specifically, for a one dimensional irreducible representation like A1g , the symmetry
coordinates will be transformed into themselves multiplied by a constant, the character
of the given symmetry operation [65]. The method for determining the symmetry coordinates for a given irreducible representation that appears in the decomposition of a reducible
representation is called the projector method. In this method, a projector is defined as
(Eqn. 3-16 in Ref. [65])
X
P (γ) ∝
χ(γ) (R)∗ OR ,
(C.5)
R
where γ is the chosen irreducible representation and {R} are all of the symmetry operations
of the group. χ(γ) (R)∗ is the complex conjugate of the character of the symmetry operation
R in the irreducible representation γ and OR is the operator that transforms a hyper-vector
into the form it would take after the symmetry operation R [65]. The proportionality symbol is introduced by the desire to have normalized and orthogonal symmetry coordinates.
108
As we are only concerned with the irreducible representation A1g , the characters of all of
the symmetry operations are 1, so Eqn. (C.5) simplifies to
X
P (A1g ) ∝
OR .
(C.6)
R
The simplest way to use P (A1g ) to determine the symmetry coordinates for A1g of Γ is
to chose one element of Γ and operate P (A1g ) on it to determine what elements of Γ are
contained in the resulting symmetry coordinate. This is repeated, choosing an element of
Γ that is not in the previous symmetry coordinate, operating P (A1g ) on this element, and
repeating this process until all of the elements of Γ are accounted for. Performing this
operation yields the following symmetry coordinates for the irreducible representation A1g
of the representation Γ, which we will refer to as {Xn }:
X1 = Jx1 Jx2 + Jy1 Jy2 + Jx1 Jx3 + Jz1 Jz3 + Jy1 Jy4 + Jz1 Jz4
+ Jy2 Jy3 + Jz2 Jz3 + Jx2 Jx4 + Jz2 Jz4 + Jx3 Jx4 + Jy3 Jy4
(C.7)
X2 = Jz1 Jz2 + Jy1 Jy3 + Jx1 Jx4 + Jx2 Jx3 + Jy2 Jy4 + Jz3 Jz4
(C.8)
X3 = Jx1 Jy2 + Jy1 Jx2 + Jx1 Jz3 + Jz1 Jx3 + Jy1 Jz4 + Jz1 Jy4
− Jy2 Jz3 − Jz2 Jy3 − Jx2 Jz4 − Jz2 Jx4 − Jx3 Jy4 − Jy3 Jx4
(C.9)
X4 = Jx1 Jz2 + Jy1 Jz2 − Jz1 Jx2 − Jz1 Jy2 + Jx1 Jy3 + Jz1 Jy3
− Jy1 Jz3 − Jy1 Jx3 + Jy1 Jx4 + Jz1 Jx4 − Jx1 Jz4 − Jx1 Jy4
+ Jx2 Jz3 + Jy2 Jz3 − Jx2 Jy3 − Jz2 Jx3 + Jx2 Jy4 + Jy2 Jz4
− Jy2 Jx4 − Jz2 Jy4 + Jx3 Jz4 + Jz3 Jy4 − Jy3 Jz4 − Jz3 Jx4 .
(C.10)
These symmetry coordinates are one form of the four symmetry allowed exchange interactions on the pyrochlore lattice. Examining {Xn }, we can see that they are identical to the
invariants defined as χn in Ref. [66] (we choose not to use χ here, as it is used to denote the
magnetic susceptibility). We can cast these symmetry coordinates in a form that allows
for the construction of the interaction matrix J (Eqn. (2.5)), by writing each of them as
an implicitly defined 12 × 12 matrix, where each term in Eqns. (C.7)-(C.10) is replaced
with +1 or −1 at the appropriate location in the matrix. The resulting matrix forms of
109
{Xn } are given by:












X1 = 























X2 = 























X3 = 











0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0

0
1
0
0
0
0
0
0
0
0
0
−1
1
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
−1
0
−1
0
0
0
0
1
0
0
0
0
0
0
0
−1
0
110
0
0
0
0
0
−1
0
0
0
−1
0
0
1
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
−1
0
0
0
0











,











(C.11)











,











0
0
1
0
0
0
−1
0
0
0
0
0
(C.12)
0
1
0
−1
0
0
0
0
0
0
0
0












,











(C.13)
and












X4 = 











0
0
0
0
0
1
0
1
0
0
−1
−1
0
0
0
0
0
1
−1
0
−1
1
0
0
0
0
0
−1
−1
0
0
1
0
1
0
0
0
0
−1
0
0
0
0
−1
1
0
1
0
0
0
−1
0
0
0
1
0
0
−1
0
1
1
1
0
0
0
0
−1
0
0
0
−1
0
0
−1
0
0
1
−1
0
0
0
0
0
1
1
0
1
−1
0
0
0
0
0
0
0
−1
0
−1
0
1
0
0
0
0
0
−1
1
0
0 −1
1
0
1
0
0
1
−1 0
0 −1
0
0
0
0
−1 1
0
0
0
0
0
0
−1
0
0
0
1
0
1
−1
0
0
0
0












.











(C.14)
A linear combination of these matrices will define all of the symmetry allowed nearest
neighbour interactions, so that the symmetry allowed form of J is
J = −J1 X1 − J2 X2 − J3 X3 − J4 X4 .
(C.15)
{Jn } = {J1 , J2 , J3 , J4 } are the coupling energies of each of the four symmetry allowed
nearest neighbour exchange terms.
C.2
Other Representations of the Symmetry Allowed
Nearest Neighbour Interactions
Now that we have determined the form of the symmetry allowed nearest-neighbour interactions, we can examine other representations of these interactions that are easier to
understand physically. As the symmetry allowed exchange interactions we found in Chapter C.1 are invariant under all of the symmetry operations of the Oh point group, any linear
combinations of these interactions will also be invariant under the symmetry operations of
Oh . This means that we are free to choose any form we want for the nearest neighbour
interactions, so long as it has four linearly independent terms that consist only of linear
combinations of the terms in {Xn }. Reference [66] gives us one possible representation
(denoted Xn ), which is in terms of the local crystal field coordinate system, defined in
111
Table 1.3 of Chapter 1.4. In Ref. [66] we are given the relationship between {Xn }, and two
terms of Xn , the other relationships are provided by Ref. [67] and are:
1
1
X1 = 2X1 + X2 + X3 + 2X4
2
2
1
1
X2 = −X1 + X2 + X3 − X4
2
2
1
X3 = 2X1 + X2 − X3 − X4
2
1
X4 = −4X1 + X2 − X3 + 2X4 .
2
(C.16)
(C.17)
(C.18)
(C.19)
The invariants {Xn } are given by (Table 1 in Ref. [66]):
1
1
1
1
1
1
X1 = − J1z J2z − J1z J3z − J1z J4z − J2z J3z − J2z J4z − J3z J4z
3
3
3 √
3
3
3
√
2 z +
2
J1 J2 + J1+ J2z −
J1z J2− + J1− J2z
X2 = −
√ 3
√3
2
2 ∗ z −
J1z J3+ + J1+ J3z −
J1 J3 + J1− J3z
−
3
√
√3
2 ∗ z +
2
J1 J4 + J1+ J4z −
J1z J4− + J1− J4z
−
√3
√3
2 ∗ z +
2
−
J2 J3 + J2+ J3z −
J2z J3− + J2− J3z
√3
√3
2
2 ∗ z −
J2z J4+ + J2+ J4z −
J2 J4 + J2− J4z
−
√3
√3
2 z +
2 z −
J3 J4 + J3+ J4z −
J3 J4 + J3− J4z
−
3
3
1 + + 1 − −
X3 = J1 J2 + J1 J2 + ∗ J1+ J3+ + J1− J3− + J1+ J4+ + ∗ J1− J4−
3
3
1
1
+ J2+ J3+ + ∗ J2− J3− + ∗ J2+ J4+ + J2− J4− + J3+ J4+ + J3− J4−
3
3
1 + − 1 − + 1 + − 1 − + 1 + − 1 − +
X4 = − J1 J2 − J1 J2 − J1 J3 − J1 J3 − J1 J4 − J1 J4
6
6
6
6
6
6
1 + − 1 − + 1 + − 1 − + 1 + − 1 − +
− J2 J3 − J2 J3 − J2 J4 − J2 J4 − J3 J4 − J3 J4 .
6
6
6
6
6
6
(C.20)
(C.21)
(C.22)
(C.23)
In this representation of the invariants, Jaα and Jbβ , where α, β = z, +, −, are expressed in
local quantization coordinates as opposed to the global cartesian coordinates used in the
112
{Xn } representation. Ja+ = Jax + iJay and Ja+ = Jax − iJay are the conventional raising and
lowering operators, and = exp (2πi/3) is a phase factor.
The conversion between various forms of the symmetry allowed nearest neighbour interactions lends itself well to matrix representation. If X is the column vector made up of
the {Xn } form of the symmetry allowed nearest-neighbour interactions, defined as



X =

X1
X2
X3
X4



,

(C.24)
and X is the column vector representation of the {Xn } form of the same interactions,
defined as


X1
 X 
 2 
(C.25)
X=
,
 X3 
X4
then we can relate the two representations via the equation
X = W X,
(C.26)
where W is a matrix with the coefficients of Eqns. C.16-C.19 in the rows. This form of the
relation between Xn and Xn is also very useful in that it allows for convenient relation of
the coupling terms of the Hamiltonian as well. If we define Hex as
Hex = −J1 X1 − J2 X2 − J3 X3 − J4 X4 ,
this can be written as
Hex = −J
T
X,
(C.27)
(C.28)
where J is the column vector made up of the elements of {Jn }, defined as



J =

J1
J2
J3
J4
113



.

(C.29)
Equivalently, we could write
Hex = −J1 X1 − J2 X2 − J3 X4 − J4 X4 ,
which has the vector form
T
Hex = −J X,
(C.30)
(C.31)
where

J1
J2
J3
J4



.

(C.32)
Hex = −J
T
X
(C.33)
= −J
T
W X
(C.34)


J =

Then, using Eqn. (C.26), we can write
T
= −J X,
(C.35)
arriving at the relationship
T
J =J
T
W,
(C.36)
which provides us with a very convenient method of converting between the coupling
energies of various representations of the symmetry allowed nearest neighbour interactions.
One particular representation that will be very useful is the “physically motivated”
representation. This representation is made of up of exchange terms commonly seen in
physics literature instead of the rather arbitrary terms in Xn and Xn . In this representation,
we define the nearest neighbour interaction Hamiltonian as
Hex = HIsing + Hiso + Hpd + HDM .
X
HIsing = −JIsing
(Jai · ẑa ) Jbj · ẑb ,
(C.37)
(C.38)
<i,j;a,b>
is the nearest neighbour local [111] Ising interaction on the pyrochlore lattice.
X
Hiso = −Jiso
Jai · Jbj ,
<i,j;a,b>
114
(C.39)
is isotropic nearest neighbour exchange, and
X
b
ab
Hpd = −Jpd
ij )(Jj · R̂ij )),
(C.40)
<i,j;a,b>
is pseudo-dipolar exchange. This interaction is physically unrelated to the long-range
magnetic dipolar interaction, but has the same mathematical form.
X
b
a
(C.41)
HDM = −JDM
Ωa,b
DM · Ji × Jj ,
<i,j;a,b>
is the Dzyaloshinskii-Moriya (DM) interaction on the pyrochlore lattice [51]. In all of these
terms, a and b are the tetrahedral sublattice sites of the pyrochlore lattice, while i and j
are FCC lattice sites. The vectors ẑa and ẑb are the local h111i directions at the a and b
ab
sublattice sites defined in Table 1.3. R̂ab
ij are unit vectors in the directions Rij , the vectors
joining the ath sublattice site of ith FCC lattice site with the bth sublattice site of the j th
FCC lattice site. The vector Ωa,b
DM is the DM vector for the pyrochlore lattice [51], which
is specified for the two pyrochlore lattice sites i, a and j, b by the five Moriya rules [68],
which are:
1. When a centre of inversion is present at the midpoint (C) of a straight line joining
two lattice sites i, a and j, b (AB), then Ωa,b
DM = 0,
2. When a mirror plane perpendicular to AB passes through the midpoint of AB, Ωa,b
DM
will be parallel to the mirror plane,
3. When there is a mirror plane that includes the lattice sites i, a and j, b, Ωa,b
DM will be
perpendicular to the mirror plane,
4. When a two-fold rotation axis perpendicular to AB passes through C, Ωa,b
DM will be
perpendicular to the two-fold rotation axis,
5. When there is an n-fold rotation axis along AB, Ωa,b
DM will be parallel to AB.
The application of all of these rules yields the Ωa,b
DM vectors shown in Fig. C.1 and defined
in Table C.3. Note that we have chosen the indirect form of the DM vectors, which differs
from the direct form only in the overall sign of the interaction, and that they are not
normalized. This has been done for ease of use, as this form of the DM vectors gives
HDM ≡ J4 X4 .
115
Table C.3: The indirect Dzyaloshinskii-Moriya vectors for the pyrochlore lattice.
1,2
ΩDM
1,3
ΩDM
1,4
ΩDM
2,3
ΩDM
2,4
ΩDM
3,4
ΩDM
(1, −1, 0)
(−1, 0, 1)
(0, 1, −1)
(0, −1, −1)
(1, 0, −1)
(−1, −1, 0)
4
3
2
z
y
1
x
Figure C.1: The indirect Dzyaloshinskii-Moriya vectors for the pyrochlore lattice [21].
116
If we recast Eqn. (C.37) in the form
Hex = −JIsing XIsing − Jiso Xiso − Jpd Xpd − JDM Xpd ,
where XIsing =
HIsing
,
−JIsing
Xiso =
vector form as Hex = −J
T
e
Hiso
,
−Jiso
Xpd =
Hpd
,
−Jpd
and XDM =
HDM
,
−JDM
(C.42)
then we can cast it in
X e . In this notation



Je = 

JIsing
Jiso
Jpd
JDM

XIsing
Xiso
Xpd
XDM



,

(C.43)
and



Xe = 



.

(C.44)
The relationship between X e and X is given by the matrix

− 13 31 − 31 13
 1 1 0 0 


U =  −1
,
 2 1 −3

0
2
0 0 0 1
(C.45)
X e = U X.
(C.46)

and the equation
This means the the coupling energies {Je } and {Jn } are related by the matrix equation
J
T
T
= J e U.
(C.47)
This relationship will be very important when presenting the results of this work. All of
the computations of the magnetic interactions in Yb2 Ti2 O7 were performed using the Xn
representation of the symmetry allowed nearest neighbour interactions because they allow
for the use of Eq. 2.4. The final Hamiltonian on the other hand is presented in the Xe
representation because it allows for easier interpretation.
117
Appendix D
Actions of the Symmetry Operations
of the Point Group Oh on the
elements of the representation Γ of
the Nearest Neighbour Exchange
Interactions on the Pyrochlore
Lattice
This appendix tabulates how the various symmetry operations of the the Oh point group
act of the elements of the representation Γ defined in Appendix C. The actions of the
symmetry operations in the first five columns of Table C.1 are given in Table D.1, while
the the actions of the rest of the symmetry operations in Oh are given in Table D.2.
These actions are described by the transformation of each of the four sub-lattice sites of
the pyrochlore lattice in to one of the other sub-lattice sites, and the transformation of
the global cartesian components under the symmetry operation. The actions of all of the
symmetry operations are key to determining the character of a given symmetry operation
in the representation Γ, and for finding the symmetry allowed exchange interactions on the
pyrochlore lattice via the symmetry coordinates of the representation.
118
119
C2
C4
C02
C3
E
Symmetry
Operation
1→1
2→2
3→3
4→4
1→1
2→3
3→4
4→2
1→2
2→3
3→1
4→4
1→1
2→2
3→4
4→3
1→3
3→2
2→4
4→1
1→2
2→1
3→4
4→3
x̂ → −x̂
ŷ → −ŷ
ẑ → ẑ
x̂ → −ŷ
ŷ → x̂
ẑ → ẑ
x̂ → −ŷ
ŷ → −x̂
ẑ → −ẑ
x̂ → −ŷ
ŷ → ẑ
ẑ → −x̂
x̂ → ẑ
ẑ → ŷ
ŷ → x̂
x̂ → x̂
ŷ → ŷ
ẑ → ẑ
1→1
2→4
4→3
3→2
1→3
3→2
2→1
4→4
1→1
2→3
3→2
4→4
1→4
4→2
2→3
3→1
1→3
3→1
2→4
4→2
x̂ → −x̂
ŷ → ŷ
ẑ → −ẑ
x̂ → ŷ
ŷ → −x̂
ẑ → ẑ
x̂ → −x̂
ŷ → −ẑ
ẑ → −ŷ
x̂ → −ẑ
ŷ → −x̂
ẑ → ŷ
x̂ → ŷ
ŷ → ẑ
ẑ → x̂
1→1
2→4
4→2
3→3
1→4
4→3
3→2
2→1
2→3
3→2
1→4
4→1
1→3
2→2
3→4
4→1
x̂ → x̂
ŷ → −ŷ
ẑ → −ẑ
x̂ → ẑ
ŷ → ŷ
ẑ → −x̂
x̂ → −ẑ
ŷ → −ŷ
ẑ → −x̂
x̂ → ŷ
ŷ → −ẑ
ẑ → −x̂
1→2
2→1
3→3
4→4
1→2
2→3
3→4
4→1
1→4
4→3
3→1
2→2
x̂ → −ẑ
ŷ → ŷ
ẑ → x̂
x̂ → ŷ
ŷ → x̂
ẑ → −ẑ
ẑ → −ŷ
ŷ → x̂
x̂ → −ẑ
1→3
2→2
3→1
4→4
1→3
3→4
4→2
2→1
1→2
2→4
3→3
4→1
x̂ → x̂
ŷ → ẑ
ẑ → −ŷ
x̂ → ẑ
ŷ → −ŷ
ẑ → x̂
x̂ → ẑ
ŷ → −x̂
ẑ → −ŷ
1→4
2→2
3→3
4→1
1→2
2→4
4→3
3→1
1→4
2→1
3→3
4→2
Actions of the Members of This Class of Symmetry Operation
(Column 1 gives the sub-lattice mappings, Column 2 gives the co-ordinate mappings)
x̂ → x̂
ŷ → −ẑ
ẑ → ŷ
x̂ → −x̂
ŷ → ẑ
ẑ → ŷ
x̂ → −ŷ
ŷ → −ẑ
ẑ → x̂
Table D.1: Actions of the symmetry operations of the point group Oh on the elements of the reducible
representation Γ.
120
σd
σh
S6
S4
i
Symmetry
Operation
1→1
2→2
3→3
4→4
1→3
3→2
2→4
4→1
1→1
2→3
3→4
4→2
1→2
2→3
3→1
4→1
1→2
2→1
3→4
4→3
1→1
2→2
3→4
4→3
x̂ → ŷ
ŷ → x̂
ẑ → ẑ
x̂ → x̂
ŷ → ŷ
ẑ → −ẑ
x̂ → ŷ
ŷ → −ẑ
ẑ → x̂
x̂ → −ẑ
ẑ → −ŷ
ŷ → −x̂
x̂ → ŷ
ŷ → −x̂
ẑ → −ẑ
1→4
4→2
2→3
3→1
1→1
2→4
4→3
3→2
1→3
3→2
2→1
4→4
1→3
3→1
2→4
4→2
1→1
2→3
3→2
4→4
x̂ → x̂
ŷ → ẑ
ẑ → ŷ
x̂ → x̂
ŷ → −ŷ
ẑ → ẑ
x̂ → ẑ
ŷ → x̂
ẑ → −ŷ
x̂ → −ŷ
ŷ → −ẑ
ẑ → −x̂
x̂ → −ŷ
ŷ → x̂
ẑ → −ẑ
2→3
3→2
1→4
4→1
1→1
2→4
3→3
4→2
1→4
4→3
3→2
2→1
1→3
2→2
3→4
4→1
x̂ → ẑ
ŷ → ŷ
ẑ → x̂
x̂ → −x̂
ŷ → ŷ
ẑ → ẑ
x̂ → −ŷ
ŷ → ẑ
ẑ → x̂
x̂ → −ẑ
ŷ → −ŷ
ẑ → x̂
1→2
2→1
3→3
4→4
1→2
2→3
3→4
4→1
1→4
4→3
3→1
2→2
x̂ → −ŷ
ŷ → −x̂
ẑ → ẑ
ẑ → ŷ
ŷ → −x̂
x̂ → ẑ
x̂ → ẑ
ŷ → −ŷ
ẑ → −x̂
1→3
2→2
3→1
4→4
1→3
3→4
4→2
2→1
1→2
2→4
3→3
4→1
x̂ → −ẑ
ŷ → ŷ
ẑ → −x̂
x̂ → −ẑ
ŷ → x̂
ẑ → ŷ
x̂ → −x̂
ŷ → −ẑ
ẑ → ŷ
1→4
2→2
3→3
4→1
1→2
2→4
4→3
3→1
1→4
2→1
3→3
4→1
Actions of the Members of This Class of Symmetry Operation
(Column 1 gives the sub-lattice mappings, Column 2 gives the co-ordinate mappings)
x̂ → −x̂
ŷ → −ŷ
ẑ → −ẑ
x̂ → x̂
ŷ → −ẑ
ẑ → −ŷ
x̂ → ŷ
ŷ → ẑ
x̂ → −x̂
x̂ → −x̂
ŷ → ẑ
ẑ → −ŷ
Table D.2: Actions of the symmetry operations of the point group Oh on the elements of the reducible
representation Γ continued.
Appendix E
Long Range Dipolar Interactions
In this Appendix, we discuss the long-range magnetic dipolar interaction term of the magnetic Hamiltonian for Yb2 Ti2 O7 . Magnetic dipolar interactions are the interactions that
arise between magnetic moments due to the interaction of one magnetic dipole moment
with the magnetic field generated by another magnetic dipole moment. When this interaction is summed over all of the magnetic moments in a system it takes the form of
Eqn. (2.12). Computing the long-range dipolar contributions to the magnetic interactions
in this work is done using the Ewald summation technique. This technique is useful, as
the dipolar interaction is infinite ranged, so the sum over i > j; a, b involves all of the spins
in the system. This sum converges very slowly, and choosing different cut-off distances for
the infinite sum corresponds to different sample configurations and macroscopic magnetic
fields [54, 60]. To avoid these convergence issues we desire a way to approximate the infinite lattice sum and the Ewald summation technique provides just such an approximation.
This summary is based heavily on Refs. [54, 60] and is provided here in the interests of
creating a self-contained thesis. We also include some comments that correct problems
discovered with the presentation and results of Ref. [60]. The goal of the Ewald summation is to split the real space sum, which is highly dependent on boundary conditions, into
two absolutely convergent sums; one in real space and one in reciprocal space [54]. One
important aspect of the form of the Ewald summation discussed here is that it yields the
a,b
Fourier transform of the long-range dipolar interaction matrix, Du,v
(q), not the real space
interaction matrix. Some of the methods discussed in this study do require the use of real
space Ewald sums, such as the mean field calculations of χa in Chapter 6 and the Monte
Carlo methods of Chapter 7, and these are discussed in Refs. [46, 47].
The first step in the Ewald summation method is to define the long-range magnetic
dipolar interactions as a matrix of the same form as the matrix J (i, j) used to describe
121
the nearest-neighbour bilinear exchange interactions in Appendix C.1. Using this form of
the interactions we can write
X
a,b
Hdip =
(E.1)
(Jai )T D (i, j) Jbj ,
hi,a;j,bi
where Jai and Jbj are the same angular momentum vectors as in Eqn. (2.4), and D
a,b
(i, j)
is a 3 × 3 sub-matrix of the 12 × 12 matrix D (i, j), with same relationship between the
two as between J
a,b
(i, j) and Eqn. (2.5).
Now, we may proceed with presenting the Ewald summation method explained in
Refs. [54, 60] in the notation of this work. To do this we start by defining the elements of
the matrix D (i, j):
v
u
ab
ab
u
v
n̂
·
R
n̂
·
R
n̂
·
n̂
ij
ij
a,b
Du,v
(i, j) =
−3
,
(E.2)
ab 5
3
|Rab
|
|R
|
ij
ij
where Rab
ij is the vector joining the i, a and j, b pyrochlore lattice sites, as defined in Chapter
1.2, and n̂u and n̂v are the global cartesian unit vectors x̂, ŷ, and ẑ. This form of D (i, j)
can be recast as
)
(
1
a,b
.
(E.3)
Du,v
(i, j) = − (n̂u · ∇x ) (n̂v · ∇x )
Rab
−
x
ij
x=0
The Fourier transform of D (i, j), D (q), can be defined in terms of its components by:
(
)
X0 e−iq·Rab
ij
a,b
,
Du,v
(q) = − (n̂u · ∇x ) (n̂v · ∇x )
(E.4)
ab
R
−
x
ij
x=0
i
where q is a wave-vector inside the first Brillouin zone, and the prime on the sum indicates
ab
a sum over Rab
ij that does not include the term where Rij = 0. The next step is to rewrite
Eqn. (E.4) with a sum over all Rab
ij :
)
(
X e−iq·Rab
ij
a,b
Du,v
(q) = −(n̂u · ∇x ) (n̂v · ∇x )
ab
R −x i
ij
u
v
+ δa,b (n̂ · ∇x ) (n̂ · ∇x )
x=0
Next, we recall the definition of a Gaussian integral [54, 60, 69]
Z ∞
1
2
2 2
=√
e−t R dt.
|R|
π 0
122
1
|x|
. (E.5)
x=0
(E.6)
This result can be used to rewrite Eqn. (E.5) as
(
)
Z ∞
X 2 ab 2
2
ab
a,b
dt √ e−iq·x
Du,v
(q) = −(n̂u ·∇x ) (n̂v ·∇x )
e−t |Rij −x| −iq·(Rij −x) π
0
x=0
i
1 + δa,b (n̂u · ∇x ) (n̂v · ∇x )
. (E.7)
|x| x=0
We next split the integral in Eqn. (E.7) into two regions [0, α], and [α, ∞) [54]. This
division of the integral leads to the real space and reciprocal space sums mentioned earlier.
The integral over the range [0, α] contributes to the real space sum, and the integral over
the range [α, ∞) contributes to the reciprocal space sum. It is important to note that the
integral over the range [α, ∞) has a divergence at Rab
ij = 0, so a careful treatment of this
term is required [54]. The two ranges of integration are controlled by α, a convergence
parameter with units of inverse distance.
Our next step is to write Eqn. (E.7) explicitly as three separate terms, which we can
then split up and consider individually:
a,b
a,b
a,b
Du,v
(q) = Aa,b
u,v (q) + Bu,v (q) + Cu,v (q) ,
(E.8)
where
)
,
e
(q) = − (n̂ · ∇x ) (n̂ · ∇x )
0
x=0
i
(
)
Z ∞
X −t2 |Rab −x|2 −iq· Rab −x 2
a,b
0
( ij ) ij
e
Bu,v
(q) = − (n̂u · ∇x ) (n̂v · ∇x )
dt √ e−iq·x
,
π
α
x=0
i
Z ∞
2 −t2 |x|2 1
a,b
u
v
−
dt √ e
.
Cu,v (q) = δa,b (n̂ · ∇x ) (n̂ · ∇x )
|x|
π
α
x=0
Aa,b
u,v
u
Z
v
α
2
dt √ e−iq·x
π
(
X
2
ab
−t2 |Rab
ij −x| −iq·(Rij −x)
(E.9)
(E.10)
(E.11)
a,b
a,b
Now we can deal with the three terms, Aa,b
u,v (q), Bu,v (q), and Cu,v (q), starting with
Aa,b
u,v (q). To deal with this term, we note that the term in {} brackets is a periodic
function of x. Using this feature of the equation, we can rewrite it as a Fourier series
following Ref. [54],
X 2 ab 2
X
ab
f (x) =
e−t |Rij −x| −iq·(Rij −x) =
gk eik·x .
(E.12)
i
Solving for gk yields
gk=G =
k
4π e−iG·(ra −rb )
F (z) ,
v |q − G|3
123
(E.13)
where G is one of the reciprocal lattice vectors of the pyrochlore lattice, defined in Chapter
1.2, v is the volume of the unit cell, and (Eqn. C13, Ref. [54])
√
Z ∞
π
2
−z 2 y 2
y sin (y) e
dy = 3 e−1/4z ,
F (z) =
(E.14)
4z
0
where z = t/|q − G|. Combining Eqn. (E.12) with Eqn. (E.13) yields the result
f (x) =
4π X e−iG·(rb −ra −x)
F (z) .
v G
|q − G|3
(E.15)
Substituting Eqn. (E.15) into Eqn. (E.9), taking the differentials, and imposing the limits
with respect to x, we obtain the result
Aa,b
u,v
Z α
4π X [n̂u · (q − G)] [n̂v · (q − G)] −iG·rab 2
√
e
dt F (t/|q − G|) .
(q) =
v
|q − G|2
π 0
(E.16)
G
Performing the integral over the range [0, α] using Eqn. (E.14), we obtain
Aa,b
u,v (q) =
4π X [n̂u · (q − G)] [n̂v · (q − G)] −|q−G|2 /4α2 −iG·rab
e
,
v G
|q − G|2
(E.17)
where the sum is over all of the reciprocal lattice vectors G.
The G = 0 term in this sum becomes non-analytic when q = 0. The G = 0 term is
4π [n̂u · (q)] [n̂v · (q)] −|q|2 /4α2
e
.
v
|q|2
(E.18)
In previous works this term in its entirety has been treated as the macroscopic magnetic
field due to the sample geometry, and has been set equal to zero because it is poorly
behaved [54, 60]. Ref. [70] explains that this is not actually the correct way to treat this
term. Instead, we add and subtract one from the exponential, to yield
4π [n̂u · (q)] [n̂v · (q)] 4π [n̂u · (q)] [n̂v · (q)] −|q|2 /4α2
+
e
−
1
v
|q|2
v
|q|2
(E.19)
According to Ref. [70], only the first term of this expansion is the macroscopic
field, which
2
2
can be set equal to zero. The second term is well behaved at q = 0 because e−|q| /4α − 1
behaves as |q|2 for small q and cancels out the 1/|q|2 in
124
[n̂u ·(q)][n̂v ·(q)]
.
|q|2
This well behaved
component of the q = 0, G = 0 term should be kept in Aa,b
u,v (q) in order to avoid problems
at q = 0 [71].
We are now finished with the term Aa,b
u,v (q), and we may move on to the other terms in
a,b
a,b
(q). The first step in treating this term is to
Du,v (q). The next term we consider is Bu,v
rearrange Eqn. (E.10) into a known integral form. By reversing the sum and integral we
obtain
Z ∞
n 2 ab 2 o X
2
a,b
u
v
0 −iq·Rab
ij
√
Bu,v (q) = − (n̂ · ∇x ) (n̂ · ∇x )
e
dt e−t |Rij −x| .
(E.20)
π
α
i
x=0
The integral in this equation can be written as a complementary error function
Z ∞
2
2
erfc(z) = √
e−x dx.
π z
(E.21)
a,b
(q) is obtained by applying the differential operators of Eqn. (E.20),
The final form of Bu,v
taking the limit x → 0, and performing the integral over t, yielding the final form of
a,b
Bu,v
(q),
X −iq·Rab
0
ab
a,b
ab
a,b
ij .
S1a,b
e
(E.22)
Bu,v
(q) =
u,v Rij − S2u,v Rij
i
The two terms in this equation are given by:
(
)
ab
−α2 |Rab
|2
ij
|
erfc
α|R
2α e
ij
ab
u
v
√
S1a,b
+
,
u,v Rij = (n̂ · n̂ )
ab 2
ab 3
π |Rij |
|Rij |
(E.23)
and
("
u
S2a,b
u,v Rij = n̂ · Rij
ab
ab
ab
n̂v · Rij
#
6α
4α3
2
ab 2
√
+√
e−α |Rij |
ab 2
ab 4
π|Rij |
π|Rij |
)
3erfc α|Rab
|
ij
+
.
ab 5
|Rij |
(E.24)
Equations
E.22-E.24 form the real space sum discussed earlier. It is useful to note that the
P 0
means
the the term Rab
ij = 0 is not included, making the real space sum analytic [54].
i
a,b
Now we may deal with the final term of Eqn. (E.8), Cu,v
(q), which contains the Rab
ij = 0
singularity. Following Ref. [54] and applying the differential operators in Eqn. (E.11), we
obtain the result
(n̂u · n̂v ) 3 (n̂u · x) (n̂v · x) a,b
a,b
a,b
S1
(x)
−
S2
(x)
,
(q) = lim δa,b −
+
+
Cu,v
u,v
u,v
x→0
|x|3
|x|5
125
(E.25)
a,b
where S1a,b
u,v (x) and S2u,v (x) are the same functions as are defined in Eqns. (E.23),(E.24),
ab
with x replacing Rij . The limit in Eqn. (E.25) is evaluated by taking series expansions
of the exponentials to O (x2 ) and the complementary error function to O (x3 ). The final
result is a constant:
4α3
a,b
(E.26)
Cu,v
(q) = − √ (n̂u · n̂v ) δa,b .
3 π
a,b
This result gives us all of the terms in Eqn. (E.8), and thus the final form of Du,v
(q) is:
4α3
a,b
Du,v
(q) = − √ (n̂u · n̂v ) δa,b
3 π
4π X [n̂u · (q − G)] [n̂v · (q − G)] −|q−G|2 /4α2 −iG·rab
+
e
v G
|q − G|2
X −iq·Rab
0
ab
a,b
ab
ij .
+
S1a,b
e
u,v Rij − S2u,v Rij
i
126
(E.27)
Appendix F
The Magnetic Form Factor of Yb3+
The magnetic form factor for Yb3+ is taken from Ref. [72]. We take the form factor F (Q)
to have the form
2
− 1 hj2 (s)i,
(F.1)
F (s) = hj0 (s)i +
gJ
|Q|
where gJ = 8/7 is landé factor for Yb3+ , s = 4πr
, where rc = 10.026 Å is the cubic unit
c
cell dimension [6], and hj0 (s)i and hj2 (s)i are given by
hj0 (s)i = A0 exp −a0 s2 + B0 exp −b0 s2 + C0 exp −c0 s2 + D0
(F.2)
hj2 (s)i = s2 A2 exp −a2 s2 + B2 exp −b2 s2 + C2 exp −c2 s2 + D2 .
(F.3)
The coefficients A0 , a0 , B0 , b0 , C0 , c0 , D0 , A2 , a2 , B2 , b2 , C2 , c2 , and D2 are given in Table
F.1 and Table F.2.
Table F.1: The hj0 (s)i magnetic form factor coefficients for Yb3+ , reproduced from
Ref. [72], Table 4.4.5.3
Ion
Yb3+
A0
0.0416
a0
16.095
B0
0.2849
b0
7.834
127
C0
0.696
c0
2.672
D0
−0.0229
Table F.2: The hj2 (s)i magnetic form factor coefficients for Yb3+ , reproduced from
Ref. [72], Table 4.4.5.7
Ion
Yb3+
A2
0.1570
a2
18.555
B2
0.8484
128
b2
6.540
C2
0.8880
c2
2.037
D2
0.0318
Appendix G
Fourier Transform of Nearest
Neighbour Interactions on the
Pyrochlore Lattice
This appendix provides the Fourier transform of the nearest neighbour exchange interaction
matrix J , J (q). To obtain this result we start from the expanded form of J , given by
Eqn. (2.5)


0
J
1,2
J
1,3
J
1,4
 2,1
2,3
2,4
 J
0 J
J

J =  3,1
3,2
3,4
 J
J
0
J

4,1
4,2
4,3
J
J
J
0
We then note that each component matrix J
form, given by
J
a,b
(q) = 2 J
a,b
a,b



.


(G.1)
will have a different Fourier transform
(1 − δab ) cos (q · (rb − ra )) ,
(G.2)
where ra are the sublattice vectors defined in Table 1.1. Applying this to all of the component matrices of J yields Eqn. (G.3).
129
130
0
2·J
1,2
· cos (q · (r1 − r2 )) 2 · J
1,3
· cos (q · (r1 − r3 )) 2 · J
1,4
· cos (q · (r1 − r4 ))



2,3
2,4
 2 · J 2,1 · cos (q · (r − r ))
0
2 · J · cos (q · (r2 − r3 )) 2 · J · cos (q · (r2 − r4 )) 


2
1
J (q) = 
.
3,4
 2 · J 3,1 · cos (q · (r − r )) 2 · J 3,2 · cos (q · (r − r ))
0
2 · J · cos (q · (r3 − r4 )) 
3
1
3
2


4,1
4,2
4,3
2 · J · cos (q · (r4 − r1 )) 2 · J · cos (q · (r4 − r2 )) 2 · J · cos (q · (r4 − r3 ))
0
(G.3)

Appendix H
Effective Spin-1/2 Hamiltonians
As discussed in Chapter 2 our model for Yb2 Ti2 O7 is closely related to an effective spin1/2 model and in this Appendix we provide more detail on this relationship. We also
perform a direct comparison to the work of Onoda on effective spin-1/2 model of Yb2 Ti2 O7 [52], which shows that our findings of abnormally large Dzyaloshinskii-Moriya (DM)
interactions in Yb2 Ti2 O7 may in fact be consistent with other models of this material. This
is a reproduction of the supplemental material of Ref. [31].
In Chapter 2, and Appendix A, we discussed the g tensor description of the ground
state doublet of the CEF, which reflects the anisotropic nature of the ground state when
written in terms of an effective spin-1/2. The exact relationship between J and ~σ is given
by
1
g ~σ ,
(H.1)
J=
2gJ
where the g tensor defined in Appendix A. Using this relationship, we may convert the
nearest-neighbour bilinear exchange Hamiltonian, Hint = HIsing +Hiso +Hpd +HDM , defined
0
0
0
0
0
in Chapter 4.2 into an effective spin-1/2 Hamiltonian Hint
= HIsing
+ Hiso
+ Hpd
+ HDM
[31].
131
In this form of the Hamiltonian (Eqns. 5-7 in Ref. [31]):
0
HIsing
=−
0
Hiso
=−
0
Hpd
JIsing
4gJ2
Jiso
4gJ2
Jpd
=− 2
4gJ
0
=−
HDM
X
(gia~σia · ẑa ) gjb~σjb · ẑb ,
(H.2)
<i,a;j,b>
X
gia~σia · gjb~σjb ,
(H.3)
b b
(gia~σia · gjb~σjb − 3(gia~σia · R̂ab
σj · R̂ab
ij )(gj ~
ij )),
(H.4)
<i,a;j,b>
X
<i,a;j,b>
JDM
4gJ2
X
a a
b b
Ωa,b
·
g
~
σ
×
g
~
σ
i i
j j .
DM
(H.5)
<i,a;j,b>
gia is the g tensor defined in the local quantization coordinates for FCC lattice site i
and sublattice site a. gia satisfies Jai = 2g1J gia~σia , where Jai and ~σia are written in the
same local quantization coordinates. This gives us an effective spin-1/2 equivalent of our
anisotropic exchange model that will be very useful for mean field and classical Monte
Carlo simulations.
0
is not the only way to express the effective spin-1/2 Hamiltonian. In a similar
Hint
manner to Appendix C.2, it is possible to create an effective spin-1/2 Hamiltonian out of
0
any linear combination of terms in Hint
. One such possibility comes from Ref. [52] (Eqn. 8
in Ref. [31]):
HO = −Jnn
nn
X
g k σ̂rz σ̂rz 0 + g ⊥ (σ̂rx σ̂rx0 + σ̂ry σ̂ry0 )
hr,r 0 i
0
0
ˆ
ˆ
ˆ
ˆ
+g
~σr · ~nr,r0 ~σr0 · ~nr,r0 − ~σr · ~nr,r0 ~σr0 · ~nr,r0
i
ˆr0 · ~nr,r0 + ~σ
ˆr · ~nr,r0 σ̂ z 0 .
+g K σ̂rz ~σ
r
q
(H.6)
In this representation x, y, and z refer to the local coordinates at each sublattice site and
ˆr , ~nr,r0 , and ~n0 0 are defined as [52]:
φr,r0 = 0, 2π/3, −2π/3 [52]. The two-vectors ~σ
r,r
ˆr = (σ̂ x , σ̂ y ) ,
~σ
r
r
(H.7)
~nr,r0 = (cos φr,r0 , − sin φr,r0 ) ,
(H.8)
~n0r,r0 = (sin φr,r0 , cos φr,r0 ) .
(H.9)
and
132
The various g α terms are defined as [52]:
√
√
9
63
g k = 1 − 8 6x − x2 − 3 6x3 + x4 ,
2
16
√
√
45
9
g ⊥ = 1 + 4 6x + x2 − 3 6x3 + x4 ,
2
16
√
√
9 4
q
2
3
g = −2 1 − 2 6x + 9x − 3 6x + x ,
4
√
√
45 2 15 √ 3 9 4
K
g = 2 2 1 + 6x − x +
6x − x .
4
4
8
(H.10)
(H.11)
(H.12)
(H.13)
x = Vpf π /Vpf σ is the ratio of two Slater-Koster parameters, representing transfer integrals
between px /py and fx(5z2 −r2 ) /fy(5z2 −r2 ) orbitals and pz and f(5z2 −3r2 )z orbitals respectively
0
[52]. The relationship between the terms in HO and Hint
are given by (Eqns. 13-16 in
Ref. [31]):
−Jnn g k = −
gk2
(−3JIsing + Jiso − 5Jpd − 4JDM ) ,
12gJ2
2
g⊥
1
1
⊥
−Jnn g = −
Jiso − Jpd + JDM ,
48gJ2
2
2
2
g⊥
7
q
−Jnn g =
Jiso + Jpd − JDM ,
24gJ2
4
√
2gk g⊥
1
K
−Jnn g = −
Jiso − Jpd + 2JDM .
12gJ2
2
(H.14)
(H.15)
(H.16)
(H.17)
By inverting these equations it is possible to express the couplings {Je } in terms of
{JO } = {−Jnn , g k , g ⊥ , g q , g K } with one free parameter that we choose to be the overall
magnitude of the interactions, −Jnn . Because of this extra degree of freedom we can only
Jpd
DM
iso
, JIsing
, and JJIsing
from the model of [52]
compute the ratios of the terms in {Je }, JJIsing
for comparison to our results in Chapter 4.2. Expressing these ratios in terms of x and
Jpd
iso
DM
plotting them yields the results shown in Fig. H.1. The ratios of JJIsing
, JIsing
, and JJIsing
for the two anisotropic exchange models determined in this study are shown in Table H.1.
Comparing the ratios in Table H.1 to those computed using the results of Ref. [52] in
Fig. H.1, we can see that for x ≈ [−∞, −0.5] and x ≈ [20, ∞], there is fair agreement in
terms of sign and magnitude between our models and that of Ref. [52]. The best agreement
Jpd
iso
DM
occurs at x ∼ −1.5. In the range x ≈ [−0.5, 10], the ratios JJIsing
, JIsing
, and JJIsing
diverge as
JIsing → 0, which happens at multiple points, as seen in Fig. H.2, making any comparison
almost impossible [31].
133
J
pd
DM
iso
, JIsing
, and JJIsing
Figure H.1: Plots of the ratios JJIsing
Slater-Koster parameters, computed from the work of Ref. [52] (solid lines) and from the
exchange couplings of Table 4.1 as shown in Table H.1, for the CEF parameterization of
Cao et al. (dashed lines). The two plots span the ranges x ≈ [−20, −0.5] and x ≈ [10, 30].
Table H.1:
J
Jiso
, pd ,
JIsing JIsing
and
JDM
JIsing
computed from the exchange couplings in Table 4.1.
Jiso
JIsing
Jpd
JIsing
JDM
JIsing
Hodges et al. [20]
Cao et al. [19]
0.279 ± 0.008
0.241 ± 0.007
−0.36 ± 0.01
−0.35 ± 0.01
−0.33 ± 0.01
−0.33 ± 0.01
134
J
pd
iso
DM
Figure H.2: Plot of the ratios JJIsing
, JIsing
, and JJIsing
Slater-Koster parameters, computed from the work of Ref. [52] (solid lines) and from the
exchange couplings of Table 4.1 as shown in Table H.1, for the CEF parameterization of
Cao et al. (dashed lines). The plot spans the range x ≈ [−0.5, 10].
In summary, if we express our nearest-neighbour bilinear exchange models for Yb2 Ti2 O7 in terms of the recent work of Onoda et al. [52, 53], we find good agreement between
our models and the work of Ref. [52]. This is particularly interesting as it perhaps provides
an independent confirmation of a large DM exchange in Yb2 Ti2 O7 , which might otherwise
be thought of as a flaw in our model.
135
Appendix I
Units of the Magnetic Susceptibility
This appendix provides a very brief explanation of the relationship between the units
of the local susceptibility in Ref. [19], the bulk susceptibility in Ref. [34], and the RPA
susceptibility of Chapter 6. Ref. [19] reports the local susceptibility in SI/MKS units of
Bohr magnetons per Tesla, µB T−1 . Ref. [34] reports the bulk susceptibility in CGS-EMU
units of emu per mole of ytterbium, emu(mol.Yb)−1 . The RPA susceptibility is computed
in natural units of inverse temperature, K−1 . This is equivalent to units of µB K−1 because
µB = 1 is used in the calculations. In order to compare the results of calculations based
on our anisotropic exchange model to experiment, we desire a way to convert the RPA
susceptibility into the appropriate experimental units.
As explained, Ref. [19] reports the local susceptibility in SI/MKS units of µB T−1 . To
relate these units to those of the RPA susceptibility (K−1 ) we define the constant CSI/MKS ,
such that χSI/MKS = CSI/MKS · χRPA . CSI/MKS is given by:
CSI/MKS =
gJ2 µB
,
kB
(I.1)
where gJ is the Landé g factor for Yb2 Ti2 O7 , µB is the Bohr magneton, and kB is the
Boltzmann constant. In SI/MKS units µB = 9.274 × 10−24 JT−1 and kB = 1.3806504 ×
10−23 JK−1 . gJ for Yb2 Ti2 O7 is 8/7. Inserting these values into Eqn. (I.1) gives the result
CSI/MKS = 0.87734 µB KT−1 . Multiplying the RPA susceptibility in units of µB K−1 by
CSI/MKS gives the susceptibility in the desired units of µB /T.
Ref. [34] reports the bulk susceptibility in CGS-EMU units of emu(mol.Yb)−1 , where
emu stands for the CGS-EMU units of magnetic susceptibility. In order to compare our
136
calculations of the bulk susceptibility to the experimental results of Ref. [34] we define the
constant CCGS−EMU , such that χCGS−EMU = CCGS−EMU · χRPA . CCGS−EMU is given by:
CCGS−EMU =
gJ2 µ2B NA
,
kB
(I.2)
where once again gJ is the Landé g factor, µB is the Bohr magneton, kB is the Boltzmann
constant, and NA is Avogadro’s number. CCGS−EMU includes an extra factor of µB compared to CSI/MKS because we are switching unit systems, so this extra µB is essentially
µB,EMU−CGS /µB,SI/MKS where µB,SI/MKS is set to one. Because we desire the magnetic susceptibility in EMU-CGS units, we must compute CCGS−EMU in CGS-EMU units, where
µB = 9.274 × 10−21 ergG−1 , NA = 6.022 × 1023 , and kB = 1.38 × 10−16 ergK−1 . Inserting
these values into Eqn. (I.2) gives C2 = 0.48999 KergG−2 mol.−1 . Multiplying the RPA
susceptibility by this value gives the desired units µB G−1 mol.−1 , the CGS-EMU units of
susceptibility per mole.
137
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